The concept of "app types" will be going away with OAuth 2.0. All apps will use the same authentication flow. Connection limits limits will be the main difference between certified (i.e. partner) and uncertified (i.e. public/private) apps. New apps will be able to connect with up to 25 organisations before they need to get certified to have the connection limit removed. Please see the OAuth 2.0 docs for more details.
All OAuth 2.0 apps will be able to maintain an offline connection (like partner apps can currently).