What's the number of data points in a Welch-based PSD?
And that's absolutely correct. As you mentioned, procedure can be described as follows:


Take long signal of length $N$ and slice it with some overlap and window into segments of length $L$.
For each of these segments calculate the DFT. Assuming no zero padding, you will get spectrum with $L$ points - same as segment length. 
Because signals are real, then we just care about first half of spectra (squred magnitude to be correct), thus take only $L/2$ points.
Do this procedure for each segment and you will end up with lot's of vectors (dependent of overlapping and signal length $N$) of length $L/2$. 
Now you need to average all the spectra. This is simply average for each frequency bin across number of segments and PSD is calculated.


Let's assume you will end up with 50 segments of length $L$, then you must average 50 spectra of length $L/2$. The longer is your signal, more segments will be averaged (better estimate), but obviously their length is not changing - you've chosen it in the beginning.