Why number of possible cuts in a graph differ from maximum number of minimum cuts?
In a specific graph, not any cut is a minimum cut. What Wikipedia claims is that out of the $2^{n-1}-1$ potential cuts, at most $\binom{n}{2}$ can be minimum cuts in any given graph. Some graphs contain even fewer minimum cuts. For example, if you connect two cycles by a single edge then there is a unique minimum cut.

The upper bound on the number of minimum cuts follows by analyzing Karger's algorithm (linked to in the question): you can show that the probability that the algorithm produces any specific minimum cut is at least $1/\binom{n}{2}$, and so there can be at most $\binom{n}{2}$ of them.