Is there a difference between the three best-reply (best-response) functions in game theory? I recent became aware that in game theory there are not one but three definitions of the best response function, which is unfortunately routinely referred to interchangeably depending on the context. People will say "the strategy is the best response" without specifying which best response they are talking about.
To showcase, let $u_i$ be the pure or mixed payoff function, $s_i \in S_i$ be the pure strategy, $x_i \in \Delta(S_i)$ the the mixed strategy of player $i$
Then the best response/reply function for player $i$ is either (depending on the context)

*

*the set of $s_i$ such that $u_i(s_i, s_{-i}) \geq u_i(s_i^\prime, s_{-i})$ for all $s_i^\prime \in S_i$


*the set of $s_i$ such that  $u_i(s_i, x_{-i}) \geq u_i(s_i^\prime, x_{-i})$ for all $s_i^\prime \in S_i$


*the set of $x_i$ such that  $u_i(x_i, x_{-i}) \geq u_i(x_i^\prime, x_{-i})$ for all $x_i^\prime \in \Delta(S_i)$
I think 1 and 3 are usually distinguished as the pure and mixed best response function. I am uncertain about $2$.
Are these concepts all distinct and give different sets, or do they in fact intersection, are subsets of one or the other? For example, is $2$ a distinct concept or equivalent to $3$? And when people refer to best response in game theory, which one are they referring to?

The set of best responses to a certain action is the set of all strategies that maximize payoff against it.
In general, it includes all actions, pure and mixed, as in 3.
For example, this is used when proving the existence of a Nash equilibrium (and then the best-response correspondence has a fixed point).
It is easy to verify that if a player has a mixed best reply, he also has a pure one so in many contexts when you just consider a certain best reply or the best reply payoff, you can assume the player uses a pure strategy, such as in 2.
Finally, in 1 you consider the best reply to a certain pure action of the others, while in 2 and 3 to a mixed. I don't see it as a difference of definition but rather of the argument of the best-reply correspondence.
