Are there specific engine TBO requirements for aircraft used in flight instruction?
§91.409   Inspections. (b) Except as provided in paragraph (c) of this section, no person may operate an aircraft carrying any person
(other than a crewmember) for hire, and no person may give flight
instruction for hire in an aircraft which that person provides,
unless within the preceding 100 hours of time in service the aircraft
has received an annual or 100-hour inspection…

You can rent out your airplane without complying with 100 hour inspections provided that you don’t provide both the pilot and the airplane. Insurance will be a lot more, probably 3-4 times as much. If you are renting it out you might have more than 100 hours in a year so you need to make sure you  comply with 100 hour ADs.
I do it with my Cherokee from time to time when I know people who want to learn to fly. The students can use any CFI they want and pay them separately from renting. Depending on how many people want to learn, I have an open pilot policy with only some CFIs and me as named insureds or if there are only a few students, everyone is a named insured.
Edit: The only maintenance requirement for Part 91 operators is 91.409 discussed above. TBO is not specifically mentioned in the FARs so you won’t find a rule saying that you don’t have to comply with manufacturer’s TBOs. I have a folder of 116 Legal Interpretations and NTSB opinions. A search of them finds no mentions of TBOs.
As you point out there are lots of articles that mention it. I'd trust them. This AVWebarticle by maintenance guru Mike Busch says,

Both Lycoming and TCM publish engine TBOs in the form of non-mandatory
service bulletins. Some Part 121/135 operators have Operations
Specifications that require them to comply with all manufacturer’s
service bulletins (even non-mandatory ones), while others have Op
Specs that require compliance only with mandatory service bulletins.