The park rental ones are 2 sizes, wide and extra wide.
A chair that is too wide is very difficult to self propel because your arms won't reach. The wheels are set farther back than would be typical for someone to self propel and they are heavy.
If you want to self people at all, I agree with renting from one of the offsite places. Even then, totally self propelling is going to be very difficult and a hard workout.
If you can walk, but need support, a rollator (rolling walker with a fold down seat) might be an option. An ECV is really the only option if you can't use a rollator and don't want to be pushed at all.
A power wheelchair with a joystick is not an option. Most of the rental companies will not rent them to people who don't have experience driving one - they are more difficult then an ECV because a single joystick controls everything - direction (forward and reverse), turning and speed all at the same time. It's quite easy to get out of control if you are not experienced.
Post 2 of the disABILITIES FAQs thread has info about renting equipment (near the top of the board or follow the link in my signature).