Yes, there will be a record of it.

You may have some challenges if your intent is to apply for citizenship, but if your timeline is 3-5 years then it will likely not be an issue as long as you don't run into other issues.

I'm not a lawyer, so please take anything I say with a healthy amount of skepticism. You should consult with an immigration lawyer for more of the intricacies of the system. Here is a discussion about this very issue that happened in 2018: https://www.justanswer.com/criminal-law/bp3ke-ticket-nj-path-train-station-saying-fare.html