Definition of a Christian: One who has accepted that a). they are a sinner
b). Jesus died for their sins as payment c). gratefully accepts the payment for that debt d). gives their life over to God, and thus ends up having a relationship with Him (meaning, they don't live with the "it's all about me and what I want" attitude anymore, but rather it's all about Him and what He wants). The last one is the tough one. People like Him to be Saviour, but they conveniently forget about the Lord part.
You can be Catholic, Baptist, Anglican, non-denominational, ect., or even not attend church like me

, and still be a Christian.
On the other hand, you can also be a Catholic, and not a Christian.
Why are they fighting? Like in Ireland you mean? Probably because they are both being pig headed

I will add though, that their are certain doctrinal Catholic traditions that cannot blend with Christianity. Like, their reverance of Mary. Respecting her, and admiring her is wonderful, but alot of Catholics pray to her, and that's bordering on the worshipping her line. And God says in the Bible that He is the only One to be worshipped.
In Catholics' defense I will add, that for example, Baptists say you must be baptised "full emersion" (meaning completely dunked) before you can be a "member" of that church. For the record, "membership" is not even in the Bible! They just want a tax break. (I was a Christian for years before I choose to be baptised anyway).
It basically comes down to, has the person met a, b, c, and d above. If yes, then they are a Christian and have a relationship with God. If no, then they are simply religious and have a faith.