Don't know if this is ever discussed, but I'm interested about this two versions of shinigami...Even Bleach was nearly rejected by Shounen Jump because there's already a shinigami related manga in it (and most probably Death Note)
Personally, I'm in the Bleach version, reasons:
1. In Death Note...ugh...they're ugly.....
2. The concepts about how a mere soul can be a shinigami (if it were like that in the real world....... )