Is c5 a key move in an opening where white begins with d4? In one game between world class players, (Alekhine and Rubinstein), the game began  1. d4 d5 2. c4 e6 3. Nf3 ...

When Rubenstein played 3... a6?, Alekhine grabbed the opportunity to reply  4. c5! because a6 was a "wasted" move and 2) weakened b6. This was the case even though it is usually not good for White to release the tension on d5 so early. (Alekhine won.)

In this game between amateurs, the game appears to have been decided by Black's 14th move 14. ... Nf8, and the fact that White could play c5 on move 18 (after exchange of several "forced" moves). I lost a game without making tactical mistakes; how can I avoid this happening in the future?

On the other hand, I was told that if Black can play c5 against d4 without adverse consequences, he equalizes. Which is why I advised Black to play 3... b6 in the game in the previous paragraph, to prepare for an eventual c5.

Is it true that in a game where White opens d4, the result of the game is largely determined by who gets to play c5?
First of all, 3...a6 after 1.d4 d5 2.c4 e6 3.Nf3 is not a terrible move for Black. It has been played by many top-level grandmasters, including Morozevich multiple times. 4.c5 is historically the eighth most common move in that position (4.cxd5 is the most common), so it's not like it refutes 3...a6. Opening theory has moved on since Alekhine.

To the more general question: putting a pawn on c5 in 1.d4 d5 positions changes the position a lot, played either as White or as Black, so it's always worth considering.

For Black, ...c5 is one of the primary ways to try to break up White's strong center based on d4. It can be easier to engineer than the other possibility, ...e5, because his dark-squared bishop is already supporting it. He can also play supporting moves like ...Nbd7, ...Qc7 (both of which are also useful for preparing ...e5), and ...Rac8. Often Black's early play is based on creating the opportunity to play ...c5; if he can get it in without problems, he's usually doing well. Note also that ...c5 is an incredibly important move in the French, again trying to pry open d4.

For White, c5 is fairly rare. It does gain space on the Queenside, but its main drawback is that it releases a lot of pressure on Black's d5 pawn. This means that his e6 pawn isn't as necessary to support it, which makes it a lot easier for him to play ...e5, since he doesn't have to worry so much about having lots of support for both the new e5 pawn and the suddenly-less-defended d5 pawn. Generally it is considered that the cons of c5 (removing pressure on d5 so that ...e5 is harder to stop) outweigh the pros (gaining space on the Queenside).

One exception is the Chebanenko Slav (1.d4 d5 2.c4 c6 3.Nf3 Nf6 4.Nc3 a6), where 5.c5 is a quite common move. One extra thing it has going for it here is that it highlights the weakness of Black's b6 square, and it's going to be a bit difficult for Black to play ...b6 to engage with it. In the game you mention, Alekhine probably had a similar idea regarding the weakness of the b6 square after 3...a6.

(18.c5 in the game you link to had nothing to do with any of this; that move is good because it entombs the bishop on b7.)