If I don't take the d pawn in the Sicilian, is it still a Sicilian?

It's not unplayable but it's not very good. cxd4 is a key idea in the Sicilian, avoiding it won't do much good. But let's see the details.
With 2... Nc6 played, d4-d5 will immediately become annoying as the knight is kicked away and therefore Black cannot really avoid taking on d4. With 2... d6 played, dxc5 will become quickly an issue. So you may have to block the d file with 2... d6 3. d4 Nd7 to avoid that. But then your only consistent plan is to go for an old Benoni setup with something like 4. d5 (you're also allowing c4 Sicilians in the process) g6 5. Nc3 Bg7 6. Bd3 Ngf6 7. 0-0 0-0. Black's plan is to play b7-b5 with a6, Rb8, Ne8-c7. It's not very healthy as a position but doesn't lose straight away and has been played many times, the b5 plan will probably force White to take action and Black can counter if White goes wrong. This version is not the best for Black though, because the addition of Nd7 is a bit clumsy, with a Benoni move-order Black would have prefered to play Na6-c7 to better prepare b5, or Bg4 to trade one pair of minor pieces to solve the space problem, or even e6 to open the e file for the Rf8.
With 2... e6 Black has a better shot at meeting both dxc5 and d5, so 3. d4 a6 has been tried in master games, it even has a name (Drazic variation) and there is some material on it. Again not very good but gives White many options he may not be ready for.
Finally there is 2. Nf3 d6 3. d4 Nf6, but it's a pure move order trick, using the fact that 4. dxc5 is not very dangerous in that situation. After 4. Nc3 Black just plays cxd4 and transpose back to the Open Sicilian, but has avoided several lines involving c4, like the Qxd4 variations with c4 and the Prins Variation (5. f3).

I saw the Drazic variation. It also has the O'Kelly Variation and a version of the St George in the French. My moves as Black are usually 1... c5, 2... e6 and if White has the Knight on f3 then on move 3. I'll play a6 or d5, adopting a French style game or a Kan/O'Kelly setup. On 2. c3 and 3. d4, I'll play d5, inviting an exchange or advance French. I also try this against the Gran Prix Sicilian. If 2.Bc4, then on the d4 advance I typically play b5 and Bb7, which sees White play d3 to hold the e pawn. I think there have to be some lines I am missing but also at the 1400-1500 elo, I noticed that players didn't play a lot of main line theory, especially the Bowlder Attack games. Still, it's worth looking at the Drazic lines. Thanks for the info.