I would use this to help students understand three "meta" ideas:

(1) Math is not about memorizing lots of random trivia. In the real world, if you go up to a mathematician and ask them which definition of a trapezoid is right, they will just smile indulgently. They don't know or care.

(2) There is not always a consensus about definitions. Get over it, boys and girls! In STEM, it's very common that when you read something, you need to check which definitions they're using.

(3) In general, in math, we prefer to make our definitions in such a way that theorems come out tidy and with a minimum of special-casing. For this purpose, it's usually good to have the things that fit definition A be a subset of the things that fit definition B. By this rule of thumb, it's preferable to define a parallelogram as being a trapezoid. If not, then any time you want to prove a theorem whose conclusion is "X is a trapezoid," you will probably have to uglify it by making the conclusion "X is either a trapezoid or a parallelogram."

Often, a reason why books will sometimes choose exclusive definitions (so that a square is not a rectangle, and a parallelogram is not a trapezoid) is that they have a low estimate of their students' intelligence. Students operating at lower intellectual levels (as well as very young kids) have trouble understanding how these definitions can be inclusive.

In this particular example, there is a possible advantage of choosing the exclusive definition, which is that then we have two sides that we can pick out as the "bases." It's a trade-off.