There is also a TER every two hours from Gare de Lyon. 

No-the name is Transilien (local trains in this region around greater Paris). Runs much more often-more like 2/hr.

All these lines have gates with magnetic striped-tickets (but changing over the chipcards). 

It IS still allowed to use passes for this-but NOT for the section In Paris itself on RER-tunnel (start gare de Lyon). You have to ask for a special ticket ´franchise´ to pass gates-not all staff will know about that. IF you succeed-ask for a-ainsi pur le retour , madame/msieur!¨ and one

Also note-do not waste passday for this-a single should be max 5€ and there are also cheapish daytickets incl all of Paris + F,,bleau metro/bus etc

It was only €5 from Gare du Nord to Fonty so I just bought that.

But actually there was no platform entry control onto the R line (transilien) from Gare de Lyon, so if the interrail card is valid there's no need for the franchise