Talk to the statisticians. They've been using "Type I error" and "Type II error" instead of "false positive" and "false negative" for ages.

In this case, though, I had much less trouble than with the statistical errors. Possibly because those are essentially the same thing, differentiated only by which hypothesis is "null". Here, though, a Type 1 system and a Type 2 system are actually very different things. Plus as others have mentioned the ordering on the systems does make sense.