There is a difference between North and South. To call both Irish is fine. To associate said word with one culture is not.
But this is the thing - nations seldom have "one culture", only in very few cases are they that homogenous. Our nations here in Britain have countless cultural entities within cultural entities, carrying their own very distinct identities, even different languages, accents, everything. Our cultures are like tapestries.
Perhaps you forget the Ulster Scots, those people that you are probably descended from? They have no cultural tie to that. I can play the bagpipes. I (Used to) wear a kilt in parades. Call it whatever you want, but I prefer when I am spoken of to be called Northern Irish or British, as that is my culture. In fact, let's add in the cultural difference of religion too. And the Huguenots, who I personally am descended from. This makes a cultural blend unique to Northern Ireland and in no way connected to the Republic of Ireland. And let's face it, honestly, when someone says Irish you do not think of bagpipes and the Ulster Scots, do you?
But shouldn't you be trying to challenge that? Can't Ulster Scots exist within a wider Irish identity in the same way that our fellow Scots that went abroad have become Scottish Americans, Scottish Canadians, Scottish Jamaicans, Scottish Argentines, Scottish Brazilians, Scottish Australians...
Of course it is quite possible. But you forget that Northerners are still a part of England. There is no border. Those vocal about being from the "North East" can be if they want. But they are still part of Scotland, and recognise that. There is no border.
But I'm trying to show you that it is possible to be as Northern Irish as you are now within a wider Irish identity. It works in Scotland and it works in England. You're getting really hung up on borders and it saddens me somewhat.
I could call myself Irish...but there is a border. There are separate cultures that are very different, with different religions, faiths and views,
There are people with different religions and faiths all over Ireland. Again, unless religion is intrinsically linked to provinces/countries as some kind of "national characteristic" I don't see what the problem is. There is nothing stopping you from being an Irish Protestant.
as well as legal systems and education systems.
I think that could be preserved in a Federal Irish state, as I advocate. I think Northern Ireland should retain all the autonomy it currently has (perhaps a bit more than they currently have, even) in a United Ireland, but it would be united with the rest of Ulster and the other Irish provinces should get the same level of autonomy as well.
"The problem is" that we are different much more drastically than any one region of Scotland is from the rest. Irish has connotations that strongly aren't Northern Irish, and therefore another term is more prudent. For example Northern Irish, or British.
Northern Ireland is very different to the rest of Ireland, but again, I don't see it as being any more different culturally speaking from the rest of Ireland as the North of England is to the rest of the UK.