There is a difference, unfortunately, between what can be called the natural grammar of English and the "formal" grammar that has been imposed upon English. You are quite correct in believing that "[t]hat was me" feels more correct than *[t]hat was I"; the "me" version follows the natural grammar of English.
However, there was a period in the past when two great circumstances came together to do what can only be described as significant damage to English, at least in formal or "standard" settings. English was, at the time, establishing itself as a language of power and influence over a vast territory, a language that would be worthy of use when discussing things that "really mattered". At the same time, one could hardly call a people educated if they couldn't read and write Latin and Greek. English had spent the last several hundred yeaars losing most of its inflections, and "respectable" languages (like Latin and Greek) were full of them, having both conjugated verbs and declined nouns. By comparison, English seemed to have very few rules, and what rules there were in English didn't transfer easily to the Classical languages.
A few scholars of note took it upon themselves to kill two birds with one stone by "fixing" the "broken" parts of English. They believed (probably correctly) that there was a sort of Universal Grammar, and were under the impression (definitely incorrectly) that this Universal Grammar would look a lot like the grammar of Latin and Greek. One of the earliest works in this vein was Hugh Jones' charmingly-titled Accidence to the English Tongue, chiefly for the use of such boys and men, as have never learnt Latin perfectly, and for the benefit of the female sex; also for the Welch, Scotch, Irish and, foreigners. Most of the blame, though, can be laid at the feet of Robert Lowth (A Short Introduction to English Grammar, written in 1762) and Lindley Murray (English Grammar Adapted to the Different Classes of Learners. With an Appendix, Containing Rules and Observations, for Assisting the More Advanced Students to Write with Perspicuity and Accuracy, written in 1795) and those who followed on from their work.
Among other things, these people decided that English should use the Latin rules for identities; one would say It is I because "it" and "I" represent the same entity (thing, person), so if "it" is the subject of the sentence, then "I" must also be treated as a subject. (That is commonly called "confusing the map with the territory". Words can represent the same thing in the real world without playing the same role in a sentence.) It is not, and never has been, a real rule in the English language, but that doesn't matter. What does matter is that it was in books written by people who were seen as authorities, and those books have guided education and the sensibilities of educated people for the past three hundred years or so.
The result is that while almost nobody says It is I in conversation, an awful lot of people will write It is I because it is what was drilled into them in school. It is almost certainly wrong by the natural rules of English, but if you lose marks for putting things the right way and all of the "respectable" books use the version that feels wrong to you, eventually you will come to accept that the sky is red no matter what your eyes tell you.