If I am not mistaken, the name given to a Returned is supposed to be based on how they died. For example, Lightsong the Bold was called such due to the way he died (I'm not sure how to do the spoiler thing, so I'll be vague). This is revealed either at some point in the book, or in the annotations on Brandon's site, I honestly cannot remember which, but do distinctly remember reading it.
It is then possible that the intent imprinted upon the Splinter is due to the final actions performed by the individual prior to their death and the bonding. It is also possible that the method for "selecting" those eligible to be Returned is that their final act is one that truly reflects the individual's values/conscience, and so the intent of the Splinter matches the individual perfectly.
So to answer your posed questions, no, I do not believe that the Splinter, nor the name given to a Returned have any effect on their will. I do not recall any Returned struggling because they want to perform a specific action, but some mysterious force is holding them back (though it has been a while since I read the book). Due to the name having to do with the manner in which the individual died, though, I think that the manner of death has to do with the eligibility to bond with a splinter. That or the individual's true nature makes them eligible, and then them acting upon that true nature when they die causes the names given to reflect the intent.
It could also, of course, simply be a literary technique employed by Sanderson to have the Returned perform actions proving their given names true.
I apologize if that felt rambly, I've never contributed to serious discussion about the books before.
