There are many ways in which you can incorrectly interpret statements on Wikipedia, which is why specialized textbooks and so forth still have use.

That's what a "false positive" is but Wikipedia also has a separate article on "false positive rate", which gives the formula

FP / (FP + TN)

Where FP is number of false positives, and TN is number of true negatives. So it's a third option:

- Out of 1000 actually negative samples, 50 were tested as positive.

So in the case of 1000 samples, 949 correctly testing as negative, 50 incorrectly testing as positive, and 1 correctly testing as positive, the false positive rate is 50 / 999.

Right; this is the definition of the numerator (the number of false positives). The false positive RATE also has a denominator, which is defined as the total number of tests performed (the second case in the parent poster's question).

Dividing by the number of positive tests instead gives what's called the 'false discovery rate' which is pretty rarely used.