The "problem" is defining what you mean by "infinite" specifically. Infinite is an adjective that you can assign to a set of numbers, and the "infinity" would be the summation of that set...

Incorrect. example: א_0 is an infinity, specifically a size of the natural number set, and is not a sum of any set. Another example: infinity in real function analysis, is a concept of unbounded growth, either positive or negative.

The set of natural numbers {1,2,3,...} can be evaluated to infinity (ℵ₀) or -1/12.

Incorrect based on previous mistake. You're describing a series sum, no series sum is 0_א, that's a major mistake, as it is specifically a set size and not a natural number.

And the 2nd concept you're referring to leads to a contradiction as a sum of positives must be positive, this means that in order to get -1/12 you must make a mistake.

These are all already well defined (except for (naive) set theory, but it's irrelevant to this), and you don't need to "define what infinity means when you use it", that's nonsense.

...? I believe I said the set of hyperreal numbers, which would contain the real numbers cross joined with the set of infinites and infinitesimals. Technically, the infinity that bounds the natural numbers would be any of those infinites. I can't really point to one specific infinity.

Infinity wouldn't be in the set of natural, rational, real, or even complex numbers. It acts as a boundary for all of those sets of numbers, but you could have a set that also includes infinity, in addition to those sets, making them the extended number set.

However, I want to point out an issue with what you said about those being different infinities... That isn't strictly true. Natural numbers, rational numbers, and powers of ten are the same level of infinite. Crazy as it is to imagine, the sets can be functionally mapped to each other. They have the same infinite of elements to them. It isn't until you add in those irrational numbers that the level of infinite increases. There is a higher order of infinite more numbers in the irrational numbers than in the rational numbers, so that infinite IS bigger.