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.