I don't understand why any private pilot would want to say they are a student pilot. I understand that talking to ATC can be intimidating when you're not used to them. It takes a lot of practice getting comfortable talking in the system. In the meantime, you just do like so many have already suggested, you say whatever you need to say, however it is handy. Right now, I'm pretty good with understanding and responding but, I will not hesitate to use my two favorite words as often as necessary, Say Again?.
BTW, one thing that speeded up my understanding and use of phraseology was listening to Live ATC. I commute long distances to work and have the app on my iPhone. I put in a busy airport such as JFK and when a controller gave instructions, I'd repeat them and wait to see what the pilot said to see if I was in the right ballpark. It takes practice.