Your definitions are not correct.
For a Strict Sense Stationary process (signal) the joint distribution of your process' value for all instants of time must be independent of time, in other words if x(t) were your process, P(x(t1),x(t2),x(t3),...) must be independent of time's origin.
For a Wide Sense Stationary process the joint probability of process value up to 2 instant of time must be independent of time, in other word P(x(t1),x(t2)) must be independent of time's origin.
Now if you have a process which at least is wide sense stationary process then your process power spectrum doesn't change by time.
To estimate a process' power spectrum from a sample signal in addition to stationarity your process also have to be ergodic, then square magnitude of Fourier transform of your signal is an estimation of power spectrum which also doesn't change with time for long enough signal.
The classic reference is stochastic processes by A. Papoulis.