I'm a fan of American English, especially of American spellings, many of which actually predate the British versions, and have said so many times on this forum. But there isn't a single good reason why "alternate" and "alternative" shouldn't retain their entirely distinct meanings. Allowing them to converge represents a degradation and a loss of nuance. On the whole, your usage of "alternate," which I believe was unconscious and unthinking, is regarded as capricious in this country. Technically, it isn't exactly incorrect as it's becoming more common in use, and it's wot people say wot's important. Of course, it's also possible that you did it deliberately in order to be perverse and contrary, but I doubt it. Anyway, I blame The Monkees for "alternate."

And I hate to tell you, but three out of four of my use of English books, which always distinguish between British and American usage, agree with me, not you, about this. The other one doesn't mention the matter at all.