Should one extract wavelet from seismic or well log for the generation of synthetic traces?
Seismic, but...

There are lots of ways of estimating wavelets. None of them rely on well logs alone, because they don't contain any information about the wavelet. Some methods are purely statistical, some use the seismic data, and some of them use seismic and well logs.

Background

I recommend reading what you can about wavelet extraction. Especially these (see links to many of these here):


Brown, A (2001). Calibrate yourself to your data! A vital first step in interpretation. Geophysical Prospecting 49, p 729–733.
Roden, R and H Sepulveda (1999). The significance of phase to the interpreter: practical guidelines for phase analysis. The Leading Edge 18, July 1999, p 774–777.
Simm, R and R White (2002), Tutorial: Phase, polarity and the interpreter's wavelet. First Break 20 (5), p 277–281.
White R, and R Simm (2003). Tutorial: Good practice in well ties. First Break 21 (10), p 75–83.


Workflow

My approach, for what it's worth, is as follows:


Establish that your seismic data are zero phase. See this wiki article and this tutorial.
Tie your wells with a purely statistical zero-phase wavelet until you know better. A Ricker is fine for most impulsive data. Don't bother with autocorrelation. Don't bother with mixed-phase. Once you've found the best one, don't change it from well to well.
Take your best well ties (maybe the best 10 or 20 percent), and check them.
Try extracting zero-phase wavelets from your best data, preferably near angle stack, and unfiltered (e.g. by fk, which processors love) &mdash; you don't want any spectral shaping nonsense. Only use the zone of interest, but at least about 1000 ms of data. Avoid the near surface.
Average all the wavelets you got and tie all your wells with this wavelet. The ties should improve in most cases.


Note: Step 4 usually involves estimating the wavelet from the seismic data, using the reflectivity from the wells. It's basically the reverse of making a synthetic: seismic / reflectivity = wavelet, where / denotes deconvolution.

Caveat interpretor

I would complete the workflow for each seismic survey (e.g. a 3D, or all lines in a 2D). Your goal is to use a single wavelet for each survey. This goal reflects the assumptions in this workflow:


The reflectivity is random.
The wavelet is zero-phase or minimum-phase.
The wavelet is stationary (invariant in space and time).


None of these assumptions hold (Ziolkowski, 1991; Why don’t we measure seismic signatures? Geophysics 56, 190–201; see also this thesis). This is what makes it fun! Your job is to cope with this without breaking the laws of physics or what we know about geology.