How to politely respond to a PhD offer follow-up email from potential supervisor?
I don't think the professor is necessarily implying that you should hurry up, they are simply putting pressure on you and hoping that you accept their offer, keeping open a channel for communication if you have questions (like a salesperson asking if you need any help as you are browsing a store), and trying to politely test whether you may have accepted another offer.

If you choose to go elsewhere, they likely want to know as soon as possible so they can extend an offer to another candidate.

I would say it is sufficient to respond with a brief, polite note that you decided before applying to wait for all offers before making a decision, and that you promise to inform them immediately once you have made a decision.


  Dr. Professor,
  
  I appreciate the email, I will certainly ask you if any questions come up. My plan when applying was to wait until I hear back from all institutions I interviewed at before making a decision.
  
  I will inform Impatient University immediately once I have made a decision, either way.
  
  Best, 
  
  Prospective Student


Your responsibility is to inform them as quickly as is reasonable once you have made a decision, but you shouldn't be pressured to respond more quickly than you are ready.

You mention "the school would offer me a small amount of fellowship (which I don't care) if I accept before late February" - this is certainly against the spirit of the April 15 Resolution, though I don't know enough about the specifics of the agreement to definitively say it's disallowed. The whole point of the resolution is to not make financial offers of any sort contingent on making a hasty decision, to keep a level playing field across institutions and in the best interest of the students. That said, I wouldn't suggest bringing this up at all or making an issue of it, unless it makes you uncomfortable in which case you could certainly consider it in your decision-making process.