Reduce only generically correct?

While it is clearly said in the Mathematica documentation that functions like FullSimplify yield only ‘generically correct’ results, I have found no mention of such restrictions for Reduce. However, consider the following two examples (Input):

Gamma[x] [Element] Reals

Gamma[0] [Element] Reals

Reduce[ForAll[x, x [Element] Reals, Gamma[x] [Element] Reals], x, Reals]

The output (Version 12.0.0) will be:

Gamma[x] [Element] Reals

False

True

of which the last output is incorrect. The second example is even more problematic, as Mathematica tells us that Not[x!=0 || y!=0] is a contradiction (Input):

expr = Implies[x == y, 1/x - 1/y == 0]

Reduce[expr, {x, y}, Reals]

% // Simplify

Reduce[Not[expr], {x, y}, Reals]

Output is:

x == y [Implies] 1/x - 1/y == 0

x < 0 || (x == 0 && (y < 0 || y > 0)) || x > 0

x != 0 || y != 0

False

Questions:

Is there a way to always get correct answers to queries such as those in the two examples above (using Reduce or any other Mathematica commands)? If not, is there a way to be always notified whenever the result is only ‘generically correct’?