Scaling of the PWELCH function in MATLAB
I believe, there are two aspects of your question.

For the FFT calculation, the power density spectrum is the square of the FFT modulus multiplied by the sampling period (in your notation: T) and divided by the number of samples (in your notation:L):
T/L * abs(FFT(X))^2  or   T/L *abs(Y(f))^2

Here is now where I believe you have an error, you have to add the power of Y(f) and (Y(-f)) and not add first Y(f) and (Y(-f) and then perform the power calculation. By adding the amplitude first, in your notation,   P1(2:end-1) = 2*P1(2:end-1)  then after calculating the power, you have the double of the correct value.

A far as the Welch method for the calculation of the Power spectrum, it takes into account all the necessary normalization. You can chose between the power spectrum (option "power") or the power density spectrum (default option "psd") 

Trying to use the option "power" for the welch calculation, it will gives as answer the amplitude of the sin squared divided by 2 (a sin as a power of 1/2): (A1^2)/2.

To calculate the power density spectrum, you have to divide the previous results by the applied frequency resolution, that is the sampling frequency by the number of samples, in your notation Fs/L. With the welch function, you have to use the option 'psd'.
Sorry for the long answer and hoping, this helps.