I have heard the term barrel trunion used in reference to many guns, but I don't have a very good idea as to what the part is specifically.

The trunion is the block that the bolt locks into and which the barrel is retained. The steel trunion allow the use of other metals to be used in the receiver, due to the trunion retaining the bolt, and not the weaker receiver.

On a HK or an AKM type rifle, the block/trunion is welded/riveted to a sheet metal receiver allowing the weapon to be produce cheaply due to the receiver not being required to be milled, nor take the force of the fired round being directly transferred to the receiver.

In regards to an AR-15, you can think of it as the locking lugs, that are attached directly to the barrel, not the receiver. The bolt, when locked up, is retained by the steel lugs/trunion, and not the weaker aluminum receiver.

With the use of a trunion, The receiver simple becomes a bolt guide and a platform for attachment such as the sights and fire control group, instead of the retention of the 50,000+ psi that is exerted when the round is fired.

So, in AR parlance the barrel extension is basically the trunion?

Yes.

Part 1 is the barrel trunion that is riveted into the receiver.  Part 3 is the sight trunion.

If you have a photo of a HK parts kit, the trunion will be the block that is attached to the barrel, used to lock up the bolt to the barrel.

WOW. Hey Troy, where did you get that picture?

CampyBoB: Is there a book I can buy that will teach me how to work on my AK variant, disassemble it, etc., just like those armorer's books I can get for my AR-15?

themao