The landing CFI demanded that the OP vacate the runway immediately, earlier than the OP intended, which is
explicitly prohibited by FAR 91.113 (g). He used his landing right-of-way to "force an aircraft off the runway surface". The OP then had to re-enter the runway to continue to his destination exit, blocking the runway a second time, and likely for a longer total duration than if he had be able to proceed as originally intended. The reg explicitly prohibits landing aircraft from forcing you to do this.
5 pages of discussion that boil down to these two sentences:
1) "
except that they shall not take advantage of this rule to force an aircraft off the runway surface which has already landed and is attempting to make way for an aircraft on final approach."
2) "except that they shall not take advantage of this rule to force an aircraft off the runway surface which has already landed and is attempting to make way, without delay at the first available taxiway, for an aircraft on final approach."
The first is FAR 91.113 (g), the second is a conglomeration of FAR 91.113 (g) regulation and AIM 4-3-20 guidance. If you choose to abide by the second sentence, and impose additional constraints beyond the regulation, do so with caution because you could find yourself in violation of FAR 91.113 (g) as the CFI in this example did.
The reg as written is simple, it's clear, and it gives the landing aircraft the right to exit the runway by their own authority. This simple wording also clearly prohibits blocking a landing aircraft by taking the runway for departure without allowing sufficient clearance as to not impact the landing aircraft.
I think it's also clear in these 5 pages that the great majority are very accommodating and will work together to achieve common goals. In the OP's scenario I also would have exited the runway ASAP as demanded by the landing aircraft, not because he told me to or that I think the regs say I had to (which they don't, it's actually the opposite), but because I consider an approaching pilot who is barking orders to other aircraft to be an unsafe pilot and I just want to get the hell away from them. The runway is no place to argue Regs, clearly that's for PoA forums
Happy flying, and landing, and best wishes in your attempts to make way.