In my brain Z is Up, Z is Height. In my job I have to deal with both all the time which is quite annoying.
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Don't forget the handedness of each coordinate system!
Minecraft
Z IS UP AND ALSO IT'S BLUE
Why would you represent Z with Y?
<i,j,k> vector master race.
I am the latter, because if I draw a X-Y plane and lay it on the ground, it aligns with that XYZ reference frame.
There are clearly more than two. In the top image that z-axis is pointing in the wrong direction.
Unless you're into magnetic resonance and use rotating frames:

uff. this is very hard.
I'm the bottom person. X and Y need to be on the same plane.
Technically, regardless of dimensions, x and y always make/share a plane.
But I agree with you, I always imagine Z as jutting out from the plane x and y make.
This reminds me of the time when I was in an industry robotics, there were a right hand rule where Z was your thumb up and X and Y your index and middle fingers. So I think the second one is the right, but it should have been drawn other way round.
x in red and z in blue please..., this is difficult to look at. My conventions !
Basically: Platformer vs Top-Down, which would you choose if you were forced to make/play a game 2D only