Status quo:

• add_variable uses 3 logical switches binary=False, continuous=False, integer=False (see MIP backends: Unify add_col, add_variable; and clarify defaulting behavior of binary, continuous, integer #20324 regarding their defaulting behavior)
• this matches the methods is_variable_continuous etc.
• set_variable_type expresses the same thing using values 1, 0, -1.

Proposal (from #20354 discussion):

• add_variable there should take a variable_type parameter, defaulting to 'continuous', with other possible values 'binary', 'integer'.
• possibly deprecate the 3 switches
• set_variable_type should take the same variable_type values 'continuous', 'binary', 'integer'.
• definitely deprecate values 1, 0, -1.

There are similar idiosyncrasies elsewhere in the backend interface (should be made separate tickets)

• Minimization is expressed as maximization=False in the constructor; as set_sense(-1); is queried by is_maximization
• ...

CC: @dimpase @videlec @jdemeyer

Component: numerical

Issue created by migration from https://trac.sagemath.org/ticket/20362