My experience is much the same as yours, giulia. Basic English writing with strict correction of spelling and sentence construction at the Primary level and then English Language as a separate subject at Secondary level. My first Grammar School loved parsing as a teaching aid and had us define each word in the sentence. When I moved areas and schools and was asked to parse a given sentence, I did as I had previously been taught, but was told that all they needed was for me to state subject and predicate.
I think we were taught the names of the tenses, because they are in the text books of the time, but I only remember that in a perfect tense the action has been completed and in a continuous tense it hasn't....maybe...
Like you I have been asked to help neighbours and friends with their English, and working out which tense is which can be tricky, as English text books produced in Spain seem to have their own nomenclature that tallies more with the Spanish names than the English. Just don't ask me to explain the Subjunctive!