Only if the loss function is C^2 (continuous Hessian), otherwise a critical point could be neither an extremal point or a saddle.

f(x) = (1 - x^2)^2

This satisfies 2) but not 3).

The point x = 0 is a critical point but it is not a global minimum and it is not a saddle point.

Could 3) be replaced by "there are no local maxima"?

3 excludes critical points from being local maxima while 2 does not, right?