Tesserarius - rank or role?

[Frank]

But then "immunis" doesn't say higher pay grade, but others seem to do so.

Thats the difference. There are tons of "Immunes" in the roman army. It is usually just a job. Most of them are simplaris, but some of them were higher payed. Just look to the ongoing dispute, if the custos armorum was a principalis or immunis and if he always was a sesquiplicarius or not.

The Tesserarius clearly shows up in roster-lists as the lowest ranked principlalis.

We should also not mix up pay rank with military rank. I lately have read an article, where an historian doubt, that every Tesserarius was a sesquiplicarius. He also doubts, that every optio and signifer was a duplicarius. And he had some good arguments. Vice versa, a miles gregarius could be a duplicarius. Some medals came with higher pay rank. And some technical staff was also payed higher without beeing a principalis.