Using rings with unity vs without unity in an algebra course

I've noticed that professors tend to make a firm decision at the beginning of their courses in Algebra about whether to consider rings as having unity or not. If they assume it does, they'll then say that a subring must have the same unity in addition to being a ring under the two operations of its containing ring.

This decision is often so pivotal that I've observed some professors choose a different textbook between years almost entirely based on whether the book treats it one way or the other, because the textbook must (I suppose) align with their choice of definition in that semester.

Main question: Why is this decision so important?

By that I mean, yes I understand that examples of what a ring/subring is or is not will be affected, but the way some Profs say it, I get the impression that this decision leads to a whole different path for the course depending on which option is chosen.

And by path, I imagine it to mean that a few important theorems down the road will be very different as a consequence of the initial choice.

Follow-up question: Is there really a very different path for a course in abstract algebra depending on this choice or is it really just an innocuous change in the concrete examples we discuss as a result of the definition? If it's the former, what are some major results that might be affected later on in the course?

Any feedback is much appreciated.

Main question: Your observations are true. But no, this decision is not important. In general, unfortunately, many people insist that their way is the unique correct way. In real life it leads to all sorts of extremism "us" vs "them", in ring theory this is $\exists 1$ vs may $\not\exists 1$, in number theory it is "include $0$ in $\Bbb N$" vs "not include $0$ in $\Bbb N$", in set theory "$A\subset U$ may mean $A=U$" vs "$A\subset B =A\subsetneq B$"; there are many more examples.

Follow-up question: If you want to treat ideals as subrings, you choose one definition, if not - another. There are other differences too. Say, Peirce decomposition has a different formulation. But all these differences are minor. If you add $\exists 1$ in the definition of a ring, you should also use the appropriate definition of a module.