Hello Marc,
Using terms “reject” and “Ho” (null hypothesis), the type I error is rejecting Ho when Ho is true. On the other hand, type II error is accepting Ho when Ho is false. 
The “false” term in false positive and false negative refers to the idea that we have committed an error.
The “positive” here is for the rejection of the null hypothesis. Once we reject the null hypothesis, we follow the alternative hypothesis which is a statement of “change” or difference. When we say false positive, we followed the statement of “change” when doing it is actually an error.
On the other hand, the “negative” term refers to “no change” hypothesis. We followed the hypothesis of “no change” (that is the null hypothesis) when in reality there is a change. Hence the term “false” because it is an error.
Hope this helps.
Best,
The 365 Team