on-chain vs off-chain bitcoin

[UnorthodoxA]

I can go and get a bitcoin through blockchains (on-chain). Also I can go to any exchange (say, Binance) and buy one (off-chain). All I know is that they are different at least with respect to confidentiality (to some extent; if one knows my wallet address on blockchain, they can follow my transactions over time but not on exchange). My question is more about the difference in nature of the two coins. I imagine the legal rights can be different.
What I get on Binance is probably just an IOU instead of an actual bitcoin. For example, is there a way to transfer that bitcoin from Binance to a ledger, without cashing out the position in Binance and buying one on-chain?
In other words, are on-chain and off-chain bitcoins (or other cryptos) identical (or fungible) animals or is there a contractual arrangement by the exchanges that make them identical (for example some type of pegging mechanism inherent in the exchange design). any reference to related articles also helps

[Upgrade00]

I can go and get a bitcoin through blockchains (on-chain). Also I can go to any exchange (say, Binance) and buy one (off-chain).

Bitcoins bought on an exchange are on chain transactions. You probably mean obtaining futures bonds or similar contracts which do not give you actual bitcoins immediately, and these are not even classified as off chain transactions. An off chain transaction would be movement of Bitcoin's across lightening network.

What I get on Binance is probably just an IOU instead of an actual bitcoin. For example, is there a way to transfer that bitcoin from Binance to a ledger, without cashing out the position in Binance and buying one on-chain?

If you do not actually own any Bitcoin in a wallet address (*to which you own the private key) then you cannot transfer to any other wallet.

[ranochigo]

Design of most exchange is such that they hold custody of your Bitcoins when you purchase and the ownership is only transferred to you when you're withdrawing from the exchange. There is no way for you to get Bitcoins on-chain without having the exchange initiate an on-chain transaction directly.

Off-chain transactions, specifically those in exchanges are tracked by the exchange themselves and is thus independent of the state on the actual blockchain; in essence, when you're purchasing Bitcoins on Binance, your account balance increases and is tracked through Binance only and nothing changes on Blockchain. It is by no means guaranteed and it is only in your custody if you decide to withdraw it.

[DireWolfM14]

If you buy bitcoin on an exchange, it doesn't move from their wallet.   The exchange just creates an internal database entry to grant you ownership of the bitcoin.  So, you are correct that there is no "on-chain" transaction in that scenario, there is only the exchange's db entry.  Not until you withdraw the coins will there be an on-chain transaction. 

The exchange itself had to obtain the coins through an on-chain transaction; either the exchange bought the coins, another customer deposited the coins into his account (on-chain transaction to the exchange's wallet,) or the exchange mined the coins.  There is no difference about the coins themselves, they just exist in a wallet that's controlled by the exchange.  

[int03h]

Centralized exchanges are designed to hold all of a user's Bitcoin. The numbers shown in your account are just numbers and only when you make a withdrawal will they approve and you get paid.
This feature is different from DEXs where you trade Bitcoin in Wrap.
When you have deposited money on centralized exchanges, it is considered that you have given the money to someone else to keep. The risk when the exchange crashes or gets hacked is that you lose your money. The Bitcoin legitimacy that you deposit into the exchange when depositing and withdrawing is equal. If you deposit dirty coins to a centralized exchange they will be confiscated.

[BlackHatCoiner]

For example, is there a way to transfer that bitcoin from Binance to a ledger, without cashing out the position in Binance and buying one on-chain?

The answer is no. If the on-chain bitcoin leaves Binance, it's not owned by them anymore. Remember, not your keys not your coins. Binance keeps your keys, that's their only power. If you own their keys, then they'll ruin their main purpose of existence. Secondly, you won't be able to trade. If you withdrew your bitcoins to your ledger, then how would it be possible to exchange your funds for other coins?

Not sure if I haven't understood your question or if you haven't understood how a hardware wallet works.

You probably mean obtaining futures bonds or similar contracts which do not give you actual bitcoins immediately, and these are not even classified as off chain transactions. An off chain transaction would be movement of Bitcoin's across lightening network.

That is true. We shouldn't name Binance's bitcoins “off-chain”. If they're off-chain, you wouldn't have to trust a third party to keep them for you.

[TangentC]

I can go and get a bitcoin through blockchains (on-chain). Also I can go to any exchange (say, Binance) and buy one (off-chain). All I know is that they are different at least with respect to confidentiality (to some extent; if one knows my wallet address on blockchain, they can follow my transactions over time but not on exchange). My question is more about the difference in nature of the two coins. I imagine the legal rights can be different.
What I get on Binance is probably just an IOU instead of an actual bitcoin. For example, is there a way to transfer that bitcoin from Binance to a ledger, without cashing out the position in Binance and buying one on-chain?
In other words, are on-chain and off-chain bitcoins (or other cryptos) identical (or fungible) animals or is there a contractual arrangement by the exchanges that make them identical (for example some type of pegging mechanism inherent in the exchange design). any reference to related articles also helps

Not familair with Binance, but let me replace it with coinbase to answer.

When you buy a Bitcoin on Coinbase , they update their internal database with your ownership of that bitcoin.
If you leave that coin there, Coinbase holds the private key , and provides the security for you and keeps backups of the private keys ,
so you don't have to worry about keeping your own backups.
That being said, if coinbase was hacked your coins could be stolen, but coinbase keeps an insurance policy on the coins in their hot trading wallets, so odds are your coins would be insured for their fiat value.

or

When you buy a Bitcoin on Coinbase , they update their internal database with your ownership of that bitcoin.
You decide to send that bitcoin to an address only you have the private keys for.
You send the bitcoin to your address (This makes a transaction on the blockchain), now it is no longer insured by coinbase,
and you are primary person responsible for making sure your private key is backed up and secured.
If you lose the private key due to fire , flood or theft/hack, you have no recourse of recovery, then your coins are just gone.

* Also if you die ,
your relatives can present coinbase with legal documents and receive your coins from the exchange.
But any coins that you did not share the private keys with your relatives would be lost forever if only you control the address.

* When you buy BTC on an exchange, you don't cash it back out to fiat to send it ,
you just send it to your BTC address, no cash out to fiat happens.
But you do have to send BTC back to an exchange before you can cash it out to fiat or another entity that is accepting BTC as payment if you want to spend it.

*Last thing, On Coinbase you can send BTC offchain to another Coinbase user at zero cost, if you send bitcoin onchain you have to pay the bitcoin transaction fees to the miners.

[Saltius]

Off-chain bitcoin is bitcoin IOU actually.
When you buy bitcoin on Coinbase, you get bitcoin IOU.
When you withdraw your fund from Coinbase to your privite wallet, Coinbase would send corresponding amount on-chain bitcoin to the address you control.

The address you control means you control the privite key to that address yourself.

The bitcoin on your address can not be confiscated by gov unless you are arrested then interrogated or even tortured to force you give out the key.
However your bitcoin IOU in Coinbase can be frozen then confiscated by gov easily.

[dkbit98]

I can go and get a bitcoin through blockchains (on-chain). Also I can go to any exchange (say, Binance) and buy one (off-chain). All I know is that they are different at least with respect to confidentiality (to some extent; if one knows my wallet address on blockchain, they can follow my transactions over time but not on exchange). My question is more about the difference in nature of the two coins. I imagine the legal rights can be different.
What I get on Binance is probably just an IOU instead of an actual bitcoin. For example, is there a way to transfer that bitcoin from Binance to a ledger, without cashing out the position in Binance and buying one on-chain?

Any purchase you make on Binance and other centralized exchanges are off-chain, and they are later offering you withdrawal in main Bitcoin chain and some on their fake chains like bsc.
They claim they are doing it to reduce withdrawal fees but it is actually all about control, centralization of power and moving funds from Bitcoin on chains they fully control in communist style.

In other words, are on-chain and off-chain bitcoins (or other cryptos) identical (or fungible) animals or is there a contractual arrangement by the exchanges that make them identical (for example some type of pegging mechanism inherent in the exchange design). any reference to related articles also helps

You can consider Lightning Network Bitcoin to be off-chain and you would pay much less fee for transactions, but I don't think Binance is supporting LN yet.
As far as I know only few exchanges like OKEX, Kraken. Bitfinex and Bitstamp are supporting Lightning Network as a way for transacting Bitcoin off-chain.