Not in this case.

FAR 119 specifically exempts flight instruction from most of the "commercial operator" rules, including the "holding out" restrictions.

The only FAR commercial requirement which applies to flight instruction is the 100 hour Mx check.

Either you or the student can provide the airplane (own or rent). If the student owns it, a 100 hour is not required. If the CFI owns it, you need the 100 hour. Rentals could go either way, depending on whether it is an FBO or club.

There are NO SPECIAL FARs which apply to part 61 flight schools. If you want to open one (large or small) you only need to comply with general part 61/91 rules for flying and whatever local/state/federal business licensing and tax laws apply. The rules for ground instruction are the usual part 61 stuff which applies to any freelance CFI.

For a part 141 flight school, there are specific operating rules which cover flight and ground instruction, and you have to apply for and achieve 141 certification from your local FSDO. You will need to actually operate as a part 61 flight school for a few years before you become eligible for 141...they don't want to give it out to amateurs. My understanding is that they are cracking down and trying to limit 141 proliferation...don't expect to see too many mom-and-pop operations getting 141 certs in the future.