Hi Carina,
I'm not providing you with right answer now, but exactly these days I have been looking for more insight on how to setup local AAA plus privilege variations, and got some useful knowledge..
I think that in your case the fundamental question is how you do access the switch (router).. I estimate you do via console, don't you? If yes, then this happens everywhere. You must define specific enable secrets for privilege levels (when other than 15), then set required user cmd set with "privilege" cmd of level not higher than your user one, and finally jumping into that level by "en X" to access it. It should work.. Actually, I wanted to test it quickly to refresh the topic, but our lab access is under maintenance at the moment.. Also to set some starting level other than 1, there should be some "privilege X" cmd attainable right under "con 0" interface.
Interesting topics discussed at :
https://learningnetwork.cisco.com/thread/32180
http://resources.intenseschool.com/ccna-security-solutions-to-facs-enable-secret-and-privilege-levels/
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EDIT after 1 hour :
My lab access got re-established, and this single command will do the job for you "aaa authorization console".
Each user in local database will jump directly to priv mode of assigned level. Same happens with VTY access (wo AAA). If AAA is used, the jump is typically assured by known trio "newmodel, aaa authen login, aaa author exec"
Hopefully I helped here and possibly earned some reward point(s) after looong time! :-D
Regards
Peter