Nils

A calculator saying error is reporting the same thing, just rudely.

The two-different-failures split is the part textbooks skip and it is exactly where the confusion lives.

The clearest way in is to remember what division is asking.

Ten divided by two asks: what number multiplied by two gives ten? Five. Division is defined as the inverse of multiplication, and the answer is whatever fills that blank.

So ten divided by zero asks: what number multiplied by zero gives ten?

And there is no such number, because anything multiplied by zero is zero. Not a hard number to find, there is nothing that satisfies the question. That is precisely what undefined means: the question has no answer, so the expression names nothing.

Now try zero divided by zero: what number multiplied by zero gives zero? Every number does. So this one has no unique answer, which is a different failure and why it gets its own name.

That is the whole thing, and notice it needs no limits, no infinity and no advanced machinery. The operation is defined by a question, and for zero the question either has no answers or has all of them.

Your infinity intuition is not silly, it is describing something real about limits, which is the next answer: but infinity is not a number that can go in that blank, so it cannot be the answer to this question.