Can I use different financial types for membership registrations and renewals without losing functionality?
As a member of the CiviCRM documentation team I will follow up on that quote, as I am not sure what it means myself. It may be left over from earlier times and no-one has been brave enough to modify or delete it. 

From experience I can tell you that you can create the new financial types for income that you need.  

My organisation also uses different accounting codes for new memberships and for renewals and I have created a different financial type for each.

For historical reasons we use a different contribution page for our renewing members than the one set up for our new members. You would not need to do that if you are prepared to have two price sets on the same page, one for new members and one for renewals.

In terms of reporting, you may need to use the line-item report from the Extended Reports Extension to give the income broken down by financial type.