Two Minimax AIs playing against each other
First of all, note that in chess, white players plays first and it is a quite complicated game to build a AI for. The fact it exists several "draw" situations makes the evaluation function even more complicated to build. You may consider practice on another strategy game like checkers.

The evaluation function:

Most of IA are indeed based on an evaluation function. Let's talk about 2 opponents strategy games without random like chess. if we neglect "draw" situation, either you have a possible winning strategy, that will make you win whatever plays the opponent, either your opponent has one. When a "draw" situation exists, one player has a possible win strategy or both have a draw strategy.

All possible combinations are nevertheless impossible to explore and finding this strategy is matter of heuristic. The evaluation fuction is an heuristic to judge if a state of the game is likely to lead to a win. It can be a value the player tries to maximize.

Generally symmetric evaluation function like yours (nb white-nb black) works well for 1vs1 games. If the two players are equivalent AI, you will indeed have

eval player 1 = - eval player 2


Your second proposition is indeed the same thing. As maximizing $A$ is like minimizing $-A$. But for a proper simulation, I would rather prefer your first proposition that treat the 2 IA independantly.

Now about the "draw" situations, a player that can hardly win (having negative eval) can decide to look for a draw instead of a win. His evaluation function may therefore not be symmetric anymore. In the same way, a player dominating the game may use an assymetric evaluation to prevent a draw.

Depth/Minimax:

I think you misunderstood what is the minimax strategy. When it is your turn to decide what to play, you will explore some possibilies. You can for instance try all your possible moves and chose the one maximizing the evaluation function. This is a depth 1 exploration.

You may also try all your moves with all the moves the opponent may do on his next turn. This is depth 2 exploration. But in all these combinations, you have no information on what your opponent will play. The minimax strategy is to assume he will play the move maximizing his evaluation function, therefore minimizing yours.

For instance, let say you have 3 possible moves $M_i$  Each leads to several possible subsequent moves $M_{i,j}$ for your opponent. $E_k$ = $evaluation(M_k)$:


$M_1$ : $E_1 = 3$,  $E_{1,j} = [4, 1]$
$M_2$ : $E_2 = 5$,  $E_{1,j} = [7, 2, -1]$
$M_3$ : $E_3 = -1$,  $E_{1,j} = [-3, 0]$


in $depth 1$, you pick $argmax_i(E_i)$, that is to say $M_2$

in $depth 2$, you pick $argmax_i(min_j(E_{i,j}))$, that is to say $M_1$

You can of course go even deeper, applying minimax strategy on each opponent action. The deeper you go, the better is the heuristic approximation to find the best strategy. But the number of states to explore generally grows as a power law of the depth ($O(a^{depth})$). Thus, all the difficulty is to focus computation on interesting branch. You can have a look to some methods:


Monte Carlo Tree Search
Pruning