I read a few discussions in this forum when experienced devs and community members said that there are ways and even services built on the Bitcoin network to speed up the confirmation time. Someone said that even instant confirmation is possible using some kind of trusted node or some other mechanism. I read those comments but never came cross with the specifics, so I am not sure what are the options and methods to speed up the confirmation time (if any).
Would you let me know please how the Bitcoin confirmation time could be speed up (even with a third party service) and if that is possible at all?

What you may be talking about is the possibility of accepting transactions as "pretty much valid" before the first confirmation since as Danny said the confirmation time cannot be changed.  This is called accepting a transaction with zero confirmations.

There are ideas floating around for this.  One example is that a third party would insure the transaction after a small amount of time but before the first confirmation.  If the transaction turned out to be bad and you did not get your BTC then the third party insurance would pay you the BTC you lost due to the fact the transaction never confirmed.  There would be a fee for this insurance and the better insurance companies would have methods to reduce their claims by studying the transaction to make sure it has a very good chance of getting confirmed.

Is that along the lines of what you are talking about?

A bad transaction is a claim on the insurance.  Nothing they can do but to pay the claim, possibly learn from the experience, and move on.  There is nothing they can do if a transaction that they thought was good turned out to be bad - probably because it was a double spend.

Point taken.

Another idea that has been talked about is that if your BTC are coming from a well known trusted source like Mt. Gox inputs.io well some sort of trusted large node then you are safe in accepting them with zero confirmation.  This hightly trusted source of BTC would publish the fact that BTC coming from a certain address or set of addresses will all be good so you can trust the transaction without waiting for a confirmation.  Perhaps they back their claims with some sort of insurance/guarantee.

True again.  So, best to wait for one or more confirmations unless you are willing to lose a certain amount of BTC to fraud.

This actually may make sense in for example a coffee shop scenario.  Most people would pay for their coffee and the transaction would eventually confirm.  If only a very small percentage of people rip you off then your losses may still be less than losses and fees on conventional credit cards - so still a win overall even accepting zero confirmation BTC transactions.

If the funds are sent from multi-sig inputs, one of which is a third party, then you can be *reasonably* sure that the level of collusion required to try and doublespend would probably not be worth it for low value transactions.

For example, there is a wallet provider called Greenaddress.it who deal in multi-sig. When I send you a payment from my wallet there both they and I must sign the tx. If I can see that the payment has originated from a (or specifically a Greedaddress.it) multi-sig address (and has a fee attached) I can be very sure that the chance of doublespend is low, i.e. I can accept zero confirmation payments with more security, no confirmation time speedup required.

Of course this is a degree of centralisation, and the merchant still requires myself and Greenaddress not to collude against them, but it is risk reduction.

This can be done without multisig too, I believe that Mycelium (android) wallet has a feature which checks recieved payments for a) a fee and b) propagation across the bitcoin network. if both of these are detected, the payment is labelled as 99% safe, even with 0 confirmations.
[Note, I have never used Mycelium wallet so my description might be a bit off]

I understand checking the fee and propagation is sufficient to look up the transaction, but such check doesn't prevent double spending or does it? I could be wrong, but I think to prevent double spending would need more validation.

It's infrequent that engineering finds a strictly superior solution, — even Bitcoin itself sacrifices many things compared to centeralized payment systems. Usually engineering is about navigating the frontier of best possible tradeoffs.  Within the Bitcoin ecosystem we can hit multiple points on that surface, as the OP notes... but asking to explain them all here is a bit too much.

For a taste of whats possible,  look up micropayment channels,  also look up green address instant transaction confirmation, look up sidechains / two-way-peg (as a more far out and general tool)...  

Green addresses are a layer on top of the bitcoin protocol, not an intrinsic part of the protocol itself.

Don't confuse the "green address" stuff discussed in 2011 with the green address instant confirmation (a feature of the green address multisig wallet service), they're different.

The original green address stuff discussed in 2011 is a horrible systemic risk that was generally rejected by the community— thank god, otherwise MTGOX's implosion this year would have resulted in tens of millions of dollars to businesses spread all around the ecosystem.