Can sleep password and screensaver password timeout settings be different? In El Capitan, is there a way to separate the request for password when the computer wakes from sleep from the request for password for stopping the screensaver?

See System Preferences --> Security &amp; Privacy --> General tab: checkbox for "Require password  after sleep or screen saver begins."

I would like to keep the password request after the computer wakes from sleep, but I don't want a password to stop the screensaver.
No this isn't possible.

Nonetheless, this would be a great general security improvment for MacOS X.
For example since I type it pretty regularly but in a safe context, I would like to have an independant password to unlock my screensaver.

I suggest you to make your question a good proposal of security improvment
on the Apple feedback web server, section about MacOS X, since they didn't yet put in place a section about security feedback.