In short, no, a priori the Catalan opening is not a gambit, because unlike the Queen's gambit, in Catalan d5 is played after white's c4, which means white is first to decide whether to take on d5 immediately or let black take on c4. Whereas, to clarify the contrast, in the Queen's gambit, 1. d4 d5 2. c4 black is the side with the tempo to decide whether or not to take on c4, thus, white is clearly sacrificing a pawn. This observation is also consistent with your examples.
In the case white leaves the matter unresolved, and allows black to take on c4 (for instance in your examples), then sure white has decided to play in the style of queen's gambit, with g3. If you wanted, you could call it a delayed queen's gambit accepted with g3, but in fact it more commonly goes by the name of Open Catalan.
Terminology discussions aside, let's now delve a bit deeper into the role of c4 in the Catalan and Open Catalan. I admit, I personally love the opening myself.
Often times names are assigned for a good reason, in the case of the Catalan, the fact that this queen's pawn opening, with g3, has a name for itself, is because the subsequent positions are very different than most sidelines of the queen's pawn with or without g3. That difference lies largely in the type of pawn structures that emerge from the Catalan. The Catalan, similar to the English opening, is very much about being conservative in the choice of pawn structure, i.e., not committing too early in the opening to a given pawn formation and keeping some options open so that ultimately, the optimal structure can be opted for purely reactively, given what black has decided to go for themselves.
Unlike the English opening in general, the Catalan, first of all, takes away the option of any early e5 by black, as after all it's a d4 opening, secondly and more importantly, it puts a special emphasis on the light square bishop, with the g3 Fianchetto, providing a natural development square for it. This is very important, white will base most of its subsequent decisions on enabling that g2 bishop as much as possible, whereas generally, in most queen's pawn variations, the light square bishop is not the central minor piece that gives a strategic direction in white's developing play, in fact there's a lot of conflict around where that bishop ought to be, e2, d3, b5, b3,..., and the conflict eventually gets resolved either purely reactively, or by another strategic decision not revolving mainly around the light square bishop (for instance, white may want to go for a certain pawn break or take away an important black piece, or cause a pawn structure damage, for instance taking the black knight on c6 when black can only recapture with the b pawn), which will decide the fate of our light square bishop. It is worth taking the time to think over this observation from a strategic point of view (start by comparing the bishop in the Catalan with the bishop in any other queen's pawn opening).
A small takeaway here is, if in the Catalan, black managed to easily force a trade off of the g2 bishop, something has probably gone wrong for white (even if slightly).
To close the loop of discussions and connect closely to your c4 question: white primarily plays c4 with the intention to challenge the long diagonal and extend the scope of the g2 bishop, even at the cost of not being able to recapture the pawn. The justification is really in the same spirit as the queen's gambit, namely, if black chooses to accept the c4 pawn, they will have permanently weakened the g2-b7 diagonal, and they will have to invest at least 2-3 tempi (a6, c6, b5,...) if they intend to keep the pawn (if not, it's easily recaptured via either Qa4, Qc2, Na3, Nbd2,...), and therein lies the compensation: white will have a huge advantage in development, which allows you to really lash out and attack black's structure from any angle you want, posing more and more problems to black and never allowing them to finish development properly, unless they accept some kind of setback.
I'm so happy to see you are looking into Kramnik's games for insights into the Catalan, he really is the go-to player when it comes to fundamental understanding of positional openings like the Catalan. So without further ado, let's take some more of his examples. Below we have two different examples of black taking on c4 and trying to hold on to it:
[Title "Vladimir Kramnik vs Anish Giri 2014 (1-0)"]
[fen ""]
1.d4 d5 2.c4 c6 3.Nf3 Nf6 4.Nc3 e6 5.g3 dxc4 6.Bg2 b5 7.Ne5 a6 8.O-O Bb7 9.b3 cxb3 10.axb3 Be7 11.Bb2 O-O 12.Qc2 Nfd7 13.Nd3 Qb6 14.Ne4 a5 15.Ndc5 Bc8 16.Qc3 b4 17.Qe3 Na6 18.Rfc1 Nc7 19.Nxd7 Bxd7 20.Nc5 Be8 21.Ra2 Qb5 22.Qd3 Qxd3 23.Nxd3 Nd5 24.Ne5 Ra6 25.Bf1 Nc3 26.Bxc3 bxc3 27.Rxc3 c5 28.dxc5 Bf6 29.f4 Bb5 30.Bg2 Ra7 31.c6 Be7 32.Be4 f6 33.Nf3 Rd8 34.e3 e5 35.fxe5 fxe5 36.Rc1 a4 37.bxa4
Notice the position at exactly move 9, black lacks so much in development, ideally, once they've castled, they'd like to play moves like Nbd7 Rc8 Qc7 and push c5. It's exemplary how Kramnik continues thereafter to exploit the weakened dark squares (irreversibly caused after black played b5/a6 to protect c4) to pressurize a5 and post a knight on c5/e5. By move 27, black has two isolated pawns both under attack by white's rook, and a knight dominating black's bishop pair, the rest is a matter of technical conversion (of the better activated and coordinated pieces). Another example:
[Title "Vladimir Kramnik vs Alexander Morozevich 2007 (1-0)"]
[fen ""]
1.Nf3 Nf6 2.c4 e6 3.g3 d5 4.d4 dxc4 5.Bg2 a6 6.Ne5 Bb4+ 7.Nc3 Nd5 8.O-O O-O 9.Qc2 b5 10.Nxd5 exd5 11.b3 c6 12.e4 f6 13.exd5 fxe5 14.bxc4 exd4 15.dxc6 Be6 16.cxb5 d3 17.c7 Qd4 18.Qa4 Nd7 19.Be3 Qd6 20.Bxa8 Rxa8 21.Bf4 Qf8 22.b6 Ne5 23.Bxe5 Qf3 24.Qd1 Qe4 25.b7 Rf8 26.c8=Q Bd5 27.f3
In this game, the critical moment is really around move 11: notice that in principle we can say both sides have quite weakened their respective queen-sides (weakened dark squares, back pawns a6, c6,...), the key difference lies on the one hand, in the comparison of the b4 bishop with the g2 bishop. The latter is directly staring at black's weaknesses, the former is staring at the irrelevant b4-e1 side-diagonal for the rest of the game. On the other hand, the story of the whole game: having provoked black into committing to a pawn structure, white has preserved options to challenge the structure in any way they want, and also to resolve their own weaknesses (as is typical for white to play b3 and a4, resolving these pawns and precluding them from becoming targets to black). From move 10 onward, comes the "lash out" phase I was speaking about, pushing b3, e4 and even sacrificing the knight!
Of course from the black side, there are a couple very different options in handling the Catalan, and by no means should the discussions above reflect that "the Catalan" is much better for white. Instead, it really comes down to style: Catalan is an extremely positional opening, where you want to get a quick development going, without having to trade off your minor pieces easily and without having to commit too early to a fixed pawn structure.
