Counting generations and other things is a matter of convention, and depends on the context. There is no right answer. But one can be precise about how one counts, or ambiguous, precision is undoubtedly preferred.
Examples:
1. Counting generations from yourself, as in your post. This could be done counting yourself as generation 1, or as generation 0. There common genealogical convention is to count generations *back* from yourself, so 1 means one generation back, i.e., your parents. If you just say "a 14th great grandparent is 12 generations," it is ambiguous because it is missing the key word "back" that lets everyone know precisely what is meant.
2. In counting generations forward from an immigrant ancestor, the convention is usually the reverse: the immigrant is called generation 1, their children are generation 2, etc. So "first generation American" means that person is themself an immigrant. Or in the well-known series of books "5th Generation Mayflower Descendants," the Mayflower Passengers themselves are generation 1.
3. In common language, 3rd cousins are people who share 2nd-great-grandparents. But in lots of Catholic records, the similar concept is "consanguinity degree." For these same cousins, their consanguinity degree would be 2, and generally that degree is one less than the number we would normally assign them as cousins.
4. In America now, people are aged 0 when they are born. But in some times and places, including earlier in American history, the standard was for someone to being life with the number "1." And the language used there was to say a person was "in the first year of their age." While this is almost unheard of these days as a way to reckon the anniversary of one's birth, it is still quite common to hear this language to refer to other anniversaries. I have heard a married couple is said to be in their 20th year of marriage, meaning they are still a little short of their "20th anniversary" ((in fact, I have been using this language around my wife for many months, since we are coming up on our 20th in July); or I've heard talk of a person who is in their 10th year of working at a particular company, meaning they have not yet received their 10 year recognition.
The important thing is to be clear about what is meant, and that is arranged by being precise with the language.
