I dont see how any of those 3 are a huge difference maker in your game. But for me? Im playing ...Bc3 doubling the pawns, creating a weakness on c3-c4, and minimizing whites queenside pawn majority. Now you have ...Na5 taking advantage of the hole on c4. You can bring your rook to c8 adding more pressure.
Your move ...Bd6 doesn't do anything, and it allows white to play f3.
You always want your moves to do as many things as possible, and for your pieces to be as active as possible.
@IMBacon
Thanks for the input. That plan makes sense. I potentially lose the center to an immediate c4 but they have to break up their bishop pair and wreck their queenside pawn majority to do it.
The pawns on c2 and c3 are likely to become targets in the long run and could result in black having a 2:1 pawn majority on the queen side.
I am not a great player. Like IMBacon I don't think any of these three moves are critically good or bad (a much stronger player may disagree). But I would probably play Bxc3 and think that in the position above I would be slightly stronger despite white having two bishops.
If you're talking about the on-board chess.com analysis tool, in general its positional evaluations can be more than a full pawn out. Never fully believe what it says unless it has a concrete, tactical line.
In this position, after bc, white must try to play c3-c4. If it weren't for ....Na5 and ...Rc8, white might still be better since against ....b5 there's counterplay.
But here black is better, as people are saying, whereas with Bd6, although it's pointing at the king, white can probably hold forever after g3.
Do listen to what Bacon is saying .... in the first diagram with unmolested pawns, white is potentially winning the ending if all the pieces come off, even though he does have two kings, giving him twice the chance of being check-mated than if he had one. 
If you're talking about the on-board chess.com analysis tool, in general its positional evaluations can be more than a full pawn out. Never fully believe what it says unless it has a concrete, tactical line.
In this position, after bc, white must try to play c3-c4. If it weren't for ....Na5 and ...Rc8, white might still be better since against ....b5 there's counterplay.
But here black is better, as people are saying, whereas with Bd6, although it's pointing at the king, white can probably hold forever after g3.
Do listen to what Bacon is saying .... in the first diagram with unmolested pawns, white is potentially winning the ending if all the pieces come off, even though he does have two kings, giving him twice the chance of being check-mated than if he had one.
Thats what i get for doing this while watching football :-)
Funny... my own list of candidate-moves to analyze would have been headed by Be7, not Bxc3 or Bd6.
Reasons?
Black's K-side activity would benefit from the Pawn advance f7-f5-f4. That requires two things... getting the Knight out of the way of the f-Pawn, and supporting the advance of the g-Pawn to g5, if required. But where can the Knight redeploy to? N(f6)-e8-d6 is one possibility... as long as you haven't put another piece on that square (such as Bd6).
Thanks @Optimissed. I looked at c3-c4 as whites next move after bxc3 and realized white was achieving one of the goals I had for black. I failed to go the next step and really look at the cost of achieving that goal. Another in a long line of examples where I stop my analysis too soon or don't analyze deep enough. I think that Bxc3 bxc3 Na5 (makes it somewhat now or never for c4) c4 Nxc4 Bxc4 dxc4 seems good for black. By stopping my analysis after bxc3 and handwaving that I helped black get the break was an error.
I also think Blueemu is right about Be7 being potentially better than Bd6.
If black plays Be7, a decent player would try to figure out why and the logical argument is that he wants to advance the f pawn. Since black is giving up a slight advantage, it makes sense to assume black is going for a bigger one. If, after Be7, white fails to pick up on the threat then next move the N is moved to e8, maintaining control over d5 and white's B is potentially in trouble since not only ...f5 but ...g5 is on the cards. So an astute white plays Bg5 after Be7 or f3 to break up white's centre. After Be7 I would be wanting to spend 20 or 30 minutes in an otb game analysing variations. For instance, is f3 better, keeping the two bishops in a more open position, which should be good for the bishops? I believe an immediate f3 is better since the e file is blocked by Be7, so black cannot so easily place pressure down that file whilst white half-opens the f-file and gets pressure and good piece activity.
So for this reason I personally think Bxc3 is the strongest bishop move. I'm more a positional player than a tactical one as I prefer to use tactics to exploit weaknesses, especially those caused by over-ambitious attacks. Perhaps there are some good tactical players who will disagree with me.
I am trying to better understand positions so that I can more easily develop a plan and identify candidate moves. This position has me confused. I see it as I have more kingside space and want to attack there, white wants to break on c4 and dissolve my center/undermine my space advantage and open the board up for white's bishop pair.
The computer loves Bxc3 almost a whole pawn better than what I played, Bd6. It also likes Ba5 as the second choice. During the game I dismissed the capture the doubled c-pawn seems to make the natural c4 pawn break easier. Bd6 feels like it is starting to use my kingside space to build something. I feel like I am fundamentally not understanding the strategy here as the computer is basically telling me to do the opposite of what I think the position is calling for. Can anyone help?