In our experience, rehearsal dinner traditions are all over the map.
When we married in 1983, our rehearsal dinner included only our attendants and we treated them to dinner at a local pub. We did invite the officiant but, because he was an order priest, he couldn't attend. The only people who needed lodging were SIL, her DH and her children. They were happily staying with MIL and FIL. We didn't invite my parents because driving into the city would have filled them with dread. No formal invitations were issued. It was just a case of, 'Let's go out and have something to eat'.
When Nephew 1 married, almost everyone had to travel. It was a rather big deal. A block of rooms had been reserved for guests and those who came were invited to the rehearsal dinner at a local Italian restaurant. Again, no formal invitations were issued. Word was passed around the hotel and everyone had a nice, informal time.
When Nephew 2 married, there was no rehearsal dinner that we knew about.
When Niece married, she and her DH hosted a rather lavish rehearsal dinner that included every guest who came in from out of town. Once again, there were no formal invitations.
Last year, we were invited to the Wedding of a rather distant relative. We sent our regrets but we did receive an invitation to the rehearsal dinner and bowling party. That invitation was sent by the parents of the Groom and included the standard RSVP card.