That's not my experience at all as a mathematician. "Maximal ideal" implicitly means "maximal proper ideal", yes, but generally ideals don't have to be proper

That's not my experience at all as a mathematician. "Maximal ideal" implicitly means "maximal proper ideal", yes, but generally ideals don't have to be proper unless specified so. If you don't include the whole ring as an ideal, you can't even define ideal addition, etc. I took an algebra class once from a professor who decided to define "ideal" to mean "proper ideal". After a few weeks he had to give it up because it just became too much trouble for reasons like that; he had to too often say "possibly improper ideal", i.e., this convention had the opposite effect he intended! I can't think of any other source I've seen use that convention.