Exactly what language do I (we) speak?

The language you speak is English; the dialect is American English (or rather, American English is a group of dialects, one of which you speak). Similarly, British English is also a dialect of English, even though it can be thought of as the "original" dialect.

Dialects are defined precisely because languages vary in different regions, be they small or large. Just as I would say soda to refer to a carbonated beverage, others might say pop, coke, or one of many other terms. That doesn't mean we speak different languages; it only means that we use different words to refer to a specific concept.

At the heart of these dialects (or families of dialects), the core of the language is the same, even though the vocabulary may vary slightly. I can easily read a book in British English and don't have to "switch" my brain over to process the words differently—I may just have to remember the small variations in each language from time to time.

The question you may ask, however, is what separates a dialect from a language and when the former becomes the latter. To that, there is no definitive answer—only what is commonly accepted and understood both by the speakers of the dialect/language in question and by linguists who have studied the history of how they have evolved.

Wikipedia offers some influencing factors, especially noting the politics of the regions. This is true, but you should always keep this in mind: "How hard is it to understand a speaker of Dialect/Language B?" Etymological factors definitely make understanding related languages easier, but note the differences between Spanish and French (both Romance languages)—and then look at American and British English.

American English and British English are considered dialects of English. These are just two of the many dialects currently in use. The definition of a dialect is:

A regional or social variety of a language distinguished by pronunciation, grammar, or vocabulary, especially a variety of speech differing from the standard literary language or speech pattern of the culture in which it exists.

The key about whether two "ways of speaking" are dialects is whether they are mutually intelligible. That is, by definition:

mutual intelligibility is recognized as a relationship between languages in which speakers of different but related languages can readily understand each other without intentional study or extraordinary effort.

Despite the vocabulary differences between British and American English, a speaker of one dialect could relatively easily understand a speaker of the other. If you read any Jane Austen, you don't need to translate the grammar (though some words may be unfamiliar). Yes, there are differences in terminology (say fanny in the UK and you'll get some embarrassed giggles) but for the most part they are the same.

Note that there are regional dialects of both American and British English which could introduce syntactic patterns which are not always mutually recognized. However, they are still part of the same language. In America, we still speak English--it just isn't the same dialect as is spoken elsewhere. If you'd like to read more about the dialects of English, you might be interested in the images of this question.

I would say (assuming you are writing in your native language), that you do indeed speak the English Language. Consider languages as like species: If two individuals can have productive intercourse using their own, then they are of the same language.

Now, there are of course multiple dialects (slightly different styles) in any language, as I'm sure you've noticed in listening to Canadians, Brits, or Aussies speak. Same goes for Southerners, New Yorkers, or New Englanders (if you aren't one of those. If you are, you probably don't notice it).

Wikipedia has a relatively complete list of dialects. If you peruse over that a bit, I suspect you may be able to pinpoint which dialect you speak. There's a good chance you speak more than one.

The difference between American English and British English is a lot smaller than the difference between different British English dialects.

American dialects are much more similar to each other, probably because most of the country was populated in a relatively short time by a homogeneous group of people. A lot of the different terms came from later immigrants, or are simply different technical terms that became popular.

By comparison a Newcastle accent had a 1000 years to develop independently of how somebody in Kent might be speaking