Yeah, that's right. Some use it for all Bush supporters, for example.

You do realize that it was a name coined by its opponents?

Yes, Harrington, but granpa Kristol & co took it on.

It is not a branch but a philosophy about international relations.

Where is the contradiction? Aren't you reducing political ideology to views on the economy here?

I consider myself followers of both schools of thought

Well, and I neither :-) My point is that in the wider world, there are more schools of thought than these too, and even the ideas of those schools can appear in different constellations.

Neoliberalism, I am still searching for its meaning and it appears to be all encompassing of what "capitalism" use to mean.

That would be a rather narrow view of views of capitalism.

Personally, I would summarize the basic ingredients of neoliberalism thus:

1. 'efficiency' is an absolute (not relative, contextual) economic quality, increasing it is desirable
   
2. no state involvement in the economy can match the 'efficiency of a market
   
3. freedom of enterprise = economic freedom
   
4. economic freedom is the most basic freedom, all other freedoms can (will) arise from it
   
5. because of the former, the political goal should be to deregulate and privatise as many functions of the state as possible, and thus increase economic freedom, even at the prize of other freedoms. (Destroy [most of] the State with the weapons of the State.)

In contrast, there are many different modes and ideologies of capitalism content with various roles for the state beyond protection of ownership rights and law enforcement, from social democracy to Gaullism. Also in contrast, even many right-libertarians have different views on how to achieve the goal (if they think about practicalities at all), while anti-corporatist, mutualist, not to mention social libertarians (rare in the US) don't think too high on any of 1-5.

_{*Lunatic*, n.
One whose delusions are out of fashion.}