1) A type I error isn't a false positive, necessarily. If you look at it from the case that the null is true then rejecting the null may be viewed as a false positive, but the alternative is another perspective in that *now* that the null is rejected, is it a correct rejection (i.e. looking at all "positives" which were incorrect) which is a different perspective.

2) Saying "assuming the null is true, I reject it erroneously" is the same as saying "I incorrectly rejected the null" except in the former I'm recognizing that I don't *know* whether the null is true or not (and the latter I'm also saying I already rejected). It's really not much of a difference because it implies "in the case of a true null, rejecting the null is incorrect and labeled as a Type I error."