There are several things to take out of this.
Firstly, as Annie has said, does it matter if your brother is "half" or "full"? It shouldn't change how you feel about him. Both of my parents were married before; I have 2 paternal half-sisters, and 1 maternal half-brother. Growing up, it didn't matter one whit to me, they were all my siblings. The only time I use the terminology is to explain to people our family dynamic.
Regarding your father claiming your brother as his own, it is possible he was the biological parent, despite the paperwork. You made mention of your father's "original" wife. Does that mean your father was married previously? If so, that could be the answer to the problem. If the marriage had not been officially severed by the time your brother was born, your father may not have wished his name on the certificate; that is provided your mother even had informed him of her pregnancy at the time.
The other scenario, which I can personally relate to myself, is that your father has met your mother and taken her as a "package deal" with her baby son. I did the same with my current wife; I helped her to raise her two children, as well as our own son. They even took my name on starting school to make matters easier. I consider them my own, and even now they still call me Dad.
Either possibility can fit in with your father's first marriage (?). That would explain why they did not marry until much later.
As Annie said, it is all very personal and much more confronting when it is now with people we know and love, rather than back in the dusty pages of history.
Darren