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My issue is about the scaling of the PSD which is output from signal.welch and signal.periodogram (as the latter is using the former), when using scaling='density' (by default). The PSD of a signal x(t) should be equal to |X(f)|^2, where X(f) is the fourier transform of x(t).

I believe that signal.welch adds a factor 2, compared to what it should be. But it is possible that it includes some extra though (only real frequencies, etc.) which is not described in enough detail for me in the documentation.

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welch and co return a onesided result when you provide a real valued time series. If you want a two sided result, you'll want to set return_onesided=False. For a one sided periodogram, the factor of 2 you have found is necessary to get the power correct. This is demoed in the example found in the docstring.