I did not use the Primary Wallet address, instead using the "Receive Bitcoins" option to the left.
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I am confused because it says on Armory that bitcoins sent to this address will show up in your Primary Wallet, so why is it not there?
I am not familiar with armory, but this may explain why armory doesn't show your balance.
Multiple wallets do check each address generated out of the seed one by one. They stop checking further addresses if they have found X addresses without a transaction history (called gap limit).
It may be possible that armory checked your first address, which turned out to have no history and then it stopped checking for further addresses, not showing your balance correctly.
In this case you still successfully received your coins and also can spend them with the corresponding private key / master seed.
I have found that the address of "Receive Bitcoins" and the address of my "Primary Wallet" are different.
This is because armory is a hierarchical deterministic wallet. This means they derive all keypairs from one seed.
Wallets do present a new address each time you want to receive a transaction for privacy reasons.
I checked blockchain.info and it shows that the transaction between the "Receive Bitcoins" address has went through.
This means you have received your btc. Everything went fine network-wise.
Is there a way to retrieve the bitcoins from the address I sent to Coinbase
You can export the private key of the address which received the funds and export it into another wallet.
But where do you want to store your coins? If you want to store them offline(?) in armory this is not necessary, since they have successfully arrived in your wallet.
If you still want to export the private key: Open the wallet properties -> backup -> other backup options -> export key list
