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This can be explained clearly in two sentences:

Division is the inverse of multiplication (a * b / b = a / b * b = a). However, multiplication by 0 is not injective (we have x * 0 = 0 for every x), so multiplication by zero can't have an inverse operation, and both 0/0 and 1/0 are undefined.

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This explanation correctly explains why 1/0 and 0/0 are not numbers, but doesn't answer the question of what makes 1/0 and 0/0 different. And they aren't both undefined.

1/0 is undefined because no value of x satisfies 1/0 = x.

0/0 is indeterminate because any value of x satisfies 0/0 = x.

0/0 is only indeterminate if your axioms allow or require it to be. Strictly speaking you can define 0/0 to mean something, but you will lose a lot of useful properties along the way. The only time 0/0 is not indeterminate is when you're dealing with something exotic and obscure like an algebraic wheel. It's correct to say 0/0 is undefined because in the vast majority of cases where someone doesn't explicitly call out the algebraic setting, they're working with a field.

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