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I am not up to date with grammatical terminology because I have learned Irish and English naturally. Most of the terminology that I have picked up I have done so here. I tend to call all "h"s in Irish séimhiús because they have the habit of mellowing letters with its distinctive séimhiú or "h" sound whether they be consonants or vowels.
ok, but as I explained in another thread, in Irish you don't lenite (séimhiú-ize) consonants in the same contexts as when you prefix h- to vowels. So if you use the word "séimhiú" anytime an h appears, it's impossible to know what you're talking about:
a chat (his cat) & a úll (his apple). Séimhiú (on cat)
a cat (her cat) & a húll (her apple). Séimhiú (on úll).
So you'd say "the possessive "a", masculine and feminine, triggers séimhiú". And the people who'll read that will think: so "a chat" means both his and her cat, and "a húll" means both his and her apple?
Many teachers say like you, they use "séimhiú" to talk about prefixing h and about putting h after a consonant, and that's why so many learners mix up everything

The way it's explained in grammars is much clearer and prevents learners from mixing all that up:
séimhiú = putting an h after the first consonant.
prefixing h to vowels = prefixing h to vowels
urú = putting one (or 2 in the case of f>bhf) consonant just before the first letter.
Like that, it's clear that there are 3 different phenomena, and that they happen in different contexts.
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What's the difference between a graphic "h" and a grammatical one?
now we're talking about Old Irish, right

Graphic h's don't exist in Modern Irish.
A graphic h is that h Old Irish scribes would put at random at the beginning of almost any word that normally begins with a vowel. Most of the time it's not pronounced, it's just to decorate or I don't know what

There's no linguistic reason for it to be there anyway

A grammatical h is that prefixed h I talked about, the one that is pronounced in certain contexts, for instance "a hech" (Old Irish, her horse). It's a grammatical h because it appears here because of a grammatical rule: feminine possessive "a" prefixes h to vowels.