Preamble: I get that we here are airing our grievances with people who 'just don't get it', and I have no issue with that. However, I think education ought to come before judgment (especially with a field as tricky as math), and I'd like to share another way of helping people to intuit the evenness of zero that has been very effective in my experience as a tutor.

A lot of people won't know the 'exists k such that...' definition of an even number, and that's totally OK. I think that one in particular is more of a 'math people' thing, as most of the other classifications you listed are way more accessible. And yes, 'can be divided by 2 to get an integer' makes perfect sense to those of us discussing on this forum, but I think we can all see how a person who's neither familiar with nor enthusiastic about mathematics could conflate this with the classic notion that 'one must not divide by zero'; does it really sound so preposterous that a layperson might mistakenly extrapolate to 'one must not divide zero'? Never mind that it's a perfectly reasonable thing to do mathematically. I think many people who aren't super comfortable with math like to 'stay on the safe side', or something.

Of course, all your criteria for evenness above are useful, but the way I've found best to explain to people intuitively is simply to show them the real number line (only marking the integers, perhaps only the nonnegatives depending on your audience), and start them with a few assumptions they're probably familiar with:

- This is a number line; it contains the positive numbers, [the negative numbers,] and zero.

- Integers (or, for the sake of explanation, 'whole numbers') alternate between positive and negative; there are not two even whole numbers next to each other on the number line.

- Zero is a number.*

Now, we just count down to 0 from 4 or someplace by even-or-oddness, which really captivates the younger students, as well as establishing the pattern in their heads that "even, odd, even, odd..." is always the case for whole numbers. When we get to zero, they fill in the blank, and so they remember as an intuitive exercise that zero has to be even. The combination of seeing where zero lives with all the other numbers on the line as well as including it in the spoken pattern is really effective. I used this all the time when I tutored elementary/middle school math, and it works pretty well from about 4th grade on.

* This idea is mightily contentious, maybe more so than the evenness. For some reason, though, there are people who will grant that zero is even, but not that it is a number. I don't know why this is the case.