"insist" means "to urge." It is used when giving a strong suggestion to someone. Using your example: "I insist on paying for the dinner" = "I would like to pay for dinner and I strongly urge that you allow me to." This is the kind of nuance that "insist" gives off. "Persist," however, is quite different. "Persist" means "to continue." If you were to say: "I persist in paying for dinner." That would probably imply that the person has payed for dinner on many occasion and continues to do so. "Persist" can also imply that the person or thing that is continuing a particular action is doing so in a very stubborn way. For example: "He persists in running in the marathon, despite his pain." = "He stubbornly continues to run in the marathon, even though he feels pain." So, as you can see, you use "insist" when expressing a desire; and persist when you want to express that something is continues. I hope this helps. Please let me know if you need further clarification.

Insist = 'I demand' or 'I really want this to happen' e.g. 'I must insist that you respect the rules of this school' Persist = to keep trying despite difficulty e.g. 'I will persist with my maths homework until I get it right'

"To insist" means to strongly demand/forcefully suggest something. In your dinner context, you are showing eagerness by using "insist". You are metaphorically pushing/forcing something (the idea of you paying for the dinner) onto someone, so it is like a strong forward movement. "To persist" means to continue (stubbornly)/to not give up. In your dinner context, persist lacks the forcefulness behind the verb "insist". It is as if you firmly plant your feet on the ground to stand strong to defend your opinion/choice. When you persist in doing something, you are not pushing anything, so the other person has room to make their own choices.

Yes, you are correct in saying "I insist on paying for dinner." It means that you will not take "no" for an answer. You are paying for the dinner. The word "persist" however means to carry on doing or saying something repeatedly usually after being refused or turned down. "My girlfriend persists in asking to share our dinner bill every time we eat out."