FYI many sources do not count the entire ring as an ideal. If you do, you’d have to define “maximal ideal” to mean “an ideal that is maximal with respect to inclusion, ignoring the entire ring ideal.”
All of algebraic geometry (a very popular branch of mathematics for studying rings) is built on the lattice of ideals. There is no way of giving a ring this structure without the ring itself being the top element of this lattice.
What you probably mean to say is that there are sources that do not treat R as a prime ideal of itself.
Just to check, I have three math textbooks from my college days that include the definition of an ideal, and none of them attempt to exclude the ring itself from the definition.
The obvious compromise is to introduce the concept of a proper ideal as an ideal that is a proper subset, and to use that when you need to exclude the ring itself. E.g., a maximal ideal is a proper ideal that is maximal with respect to inclusion.
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.