The Caro-Kann is higly strategic, more than tactic. An experienced GM like Petrosian will try this so to give more weight to his experience, trying not to get into a tactic melée, where Fisher's genius could shine (just supposing!)

If white plays c4 now, they get into a heavy, slow game.

c4 is somewhat sharp. It's a variation of the Panov-Botvinnik attack.

Book.

Wow, this is Fisher splatting at Petrosian's face "I don't care about d4", kicking the N. I'm not good enough in chess theory to tell what was white's plan in that case, but I wouldn't have allowed black to play this! Even without the pin, black seems to be able to play d4, and white... Ne2, blocking is own B... it looks terrible. But of course, I'm not even close to the half of Fisher!

Tal-Botvinnik (3rd game) 5 gxf3

Due to the Q, e4 isn't really threatened. But d5 is still on the air...

GM's moves are fare beyong my understanding. e5 seemed very good, threatening d5 and preventing Wdsb's access to f4.

In this position, it seems that bxc3, keeping the B pair and opening the column to the weak b7 would have served white very well. WK is going to g1 anyway.

Was this the better moment to play this? I don't think so, this only gets an exchange. Bd6 in the first place could pressure white's weakened kingside, flanked by Qc7. It also prevents e5 by white. But who am I to say that?  

Okay, so I'm trying to examine Black's logic here, so that we can pinpoint why he had to play subtle the way he did. Earlier, we saw ...e6, instead of e5. This is because the Black dark bishop needed to come out to pin the knight, from falling back to e2, attacking any attempt to play d4 and force an e5 off of its square. Once the bishop blocks, d4 is then safe, and if the knight doesn't retreat, then White loses castling. White instead has to lose some tempo by playing Nb1- allowing ...e5. So Petrosian always had a plan to play e5, but just in a way to allow less counterplay, especially from the knight.

...BxB, KxB, losing the castling right.

What was Petrosian's idea here? I would have understood Qb6, instead, where white can't exchange the B (BxB, QxB+, Qxb2).

Hmm... Not sure if Qb6 is entirely accurate. I think you give White a chance to break your pin via c3, and if dxc3, then Nxc3, bxc3, and Bxc3 all look good to me.

This was undoubtely Petrosian's forseen position. At the cost of both bishops, black has "blinded" white's one. Still, as black, I would have gone for Qb6. e5 is dubious from white, since the BN has a good square on d5.