This could go on for a long time... 🙂
a ball in a Euclidean space is compact (under the induced topology) and there's no reason for the definition of a ball in 1-dimensional Euclidean space to be different from the definition in higher dimensions
The balls are no longer compact in infinite dimensional spaces (where many interesting applications, e.g. quantum mechanics, are) unless you use very special (weak) topologies. Compact sets are important precisely because they are so special, not general. So I don't think compactness is a good argument. Much of function theory is done on open sets to avoid having to deal with boundaries, which is probably where my preference for open sets comes from. But I do agree with you notation for discrete sets (which are always both open and closed under the discrete topology anyway).
I think (and hope) that I have made my point, which is not that my intuition is "better" than yours, just that you could argue either way, depending on context and/or background.