Without going into your specific line, that's the general consensus, that black should rather give back the extra pawn and pursue other aims (modern play) instead of doggedly holding on to it (classical play) so that white can't get anything out of it.
A game coming out of the Slav (with a late cxd4 by black) comes to mind.. Denker - Botvinnik from the 'radio match' in 1930, where Botvinnik introduces the Botvinnik system or Anti-Meran gambit. To date, people still haven't a clue what's going on and computers aren't faring any better

To be accepted or declined during a game means to have your oppenent or yourself either accepting either an exange or a gambit,often either making that exchange or taking a pawn or other piece in a gambit is referred to as "accepted" whereas not making an exange or not taking a gambit pawn or other piece is referred to as "declined".

I read "Fire on Board" by Alexei Shirov, and it contained a section on many of his Botvinnik System games. After 1. d4 d5 2. c4 c6 3. Nc3 Nf6 4. Nf3 e6 5. Bg5 dxc4 6. e4 b5 7. e5 h6 8. Bh4 g5 9. Nxg5 hxg5 10. Bxg5 Nbd7 11. exf6, Black is down a pawn. So by accepting the gambit, black lost a pawn...?

Octal some of blacks defences or gambits often involve black losing a pawn in order to shift initiative and/or acquire positional superiority. A prime example would be the Danish gambit.

I'm not refuting that, instead I have a question: isn't the point of a gambit to sacrifice material for position? So it would be unlikely that White would change his course to pursuing material supremacy instead right? If he did, what would be so lucrative to the extent that he would think of doing so?

The Queen's Gambit isn't truly a gambit. It's just assumed that white WILL regain the gambit pawn, and it would be dumb for black to try and prevent that.

The QG is just one exception to the naming rules. Another is The Two Knights Defense, which IS actually a true gambit. After 1. e4 e5 2. Nf3 Nc6 3. Bc4 Nf6 4. Ng5 d5 5. exd5, white is up a pawn, but black gets the initiative for it.

--Fromper

In a gambit, if any material is offered, if the opponent accepts the material, then the gambit is 'accepted'; otherwise, the gambit is declined.
Classic example: the Queen's Gambit 1. d4 d5 2. d4. If Black plays 2... dxc4, then the opening is the Queen's Gambit Accepted. If Black instead plays 2... e6, then the opening is the Queen's Gambit Declined. If Black plays something else, then it is a completely different opening. Does that help?

Not always. Specific to an opening gambit, the material is usually sacrificed to gain time and initiative, and attacking chances along with it. A sacrifice by itself is more broad, and is trading the material imbalance for another imbalance, say position, or the initiative.

I hate being the last poster on a dead thread (or at least a thread on life support for five months)

After 3.e4 in the QGA, Black's best moves are 3...e5, 3...Nc6 and 3...Nf6 - personally I like 3...e5 best of all (it is also favoured by Rublevsky), but all three options provide Black with a satisfactory game. 3...b5 is playable as well, but White should emerge with a slight edge.

Honestly, 3. e3 is better for White, as Black will lose more than a pawn if he tries to hold onto it. 3. e3 b5? 4. a4 c6 axb5 cxb5 Qf3! and White wins at least a Bishop.

As for your suggested line, yes, White does the pawn back in this line. I hadn't seen this line before, to be honest.