The word billion derives from the Latin prefix 'bi' (meaning two or twice, like in bicycle, binary, biped, biplane or biennial), and the termination of million. The logically consistent meaning for billion is therefore million^2, that is, 10^12.
If this explanation is not enough for you, perhaps because you were taught that a billion is 1000*1000^2 and you didn't see the trick, read how names are assigned to numbers:
One names small numbers in terms of units. To name the number beyond 9 a new name is required, ten.
Then one names numbers in terms of number of tens plus the remainder. To name the number beyond 99 a new name is required, a hundred.
Then one names numbers in terms of number of hundreds plus the remainder of tens plus the remainder. To name the number beyond 999 a new name is required, a thousand.
Then one follows with tens of thousands and hundreds of thousands, but beyond 999 999 a new name is required, a million.
Then one follows with tens of millions, hundreds of millions, thousands of millions, tens of thousands of millions and hundreds of thousands of millions, but beyond 999 999 999 999 a new name is required, a billion.
Then one follows with tens of billions, hundreds of billions, thousands of billions, tens of thousands of billions and hundreds of thousands of billions, but beyond 999 999 999 999 999 999 a new name is required, a trillion.
And so on.
Milliards, billiards and the like are never needed. One may simply use thousand millions, thousand billions, etc.
From my point of view, using billion to mean thousand million is pure laziness. See this quote from the Oxford dictionary:
Once the business world and the financial press found themselves discussing 'thousand millions' so much, the 'American' system simply became more convenient, despite a certain lack of logical tidiness. (A 'British' trillion is the third power of a million, while the 'American' one is the fourth power of a thousand, and the 'American' system continues out of sync with the arithmetic).
NOTE: "Usamerican" is not (at least for me) a pejorative term; it is simply the most accurate term I know off to mean "people from the USA". Many US residents aren't US citizens, Canadians and Mexicans are also North Americans, and everybody in the continent are Americans. If you know of a more accurate term, please tell me.