Actually, neither A nor B in the example are a correct interpretation of a false positive rate, because there isn't enough information provided to determine a false positive rate. In the terminology of clinical diagnostic testing evaluation, testing outcomes can be classified into one of four fundamental categories: True positive, false positive, true negative, or false negative.
While there are four possible outcomes, any given patient only has two possible test results, positive or negative. And the assessment of that result depends on whether they have the disease or not. So patients with the disease can only have a true positive or a false negative, and patients without the disease can only have a false positive or a true negative.
Since it's not possible for a patient with the disease to have a false positive, we get to ignore them in calculating a false positive rate. So a false positive rate would be the number of false positives out of the total number of possible false positives (or the percentage of patients without the disease who test positive). With an ideal data set in which you knew both the test result and true infection status of each patient, you could calculate it like this:
$$ FPR = \frac {False\ Positives}{False\ Positives + True\ Negatives}$$
The thing is, we don't usually use this in evaluating clinical diagnostics, but rather it's complementary value, which is called "specificity" (for some reason), and is essentially true negative rate.
$$ Specificity = \frac {True\ Negatives}{False\ Positives + True\ Negatives}$$
But note that neither of these values can give you the number of false positives in your example without more information. If you assumed that all 100,000 people tested did not have the disease, then you'd expect 1000 false positives (given your 1% false positive rate). But it gets more complicated when you have a non-zero rate of infection in the tested population. The greater the underlying infection rate, the fewer false positives you expect to see (and vice versa, with a lower infection rate, you expect a greater proportion of positives to be false, which I think is pointed out in the article you linked). Basically, it's really hard to determine something like a false positive rate on the fly and even harder to estimate it's impact in the real word, especially when the test you're trying to evaluate is also the best means you have for determining true infection.
Another consideration is that, for a well designed PCR diagnostic test (most COVID tests are), the majority of false positive results will come from technical errors (i.e. cross contamination, mislabeled specimens, etc.), rather than randomly from the test itself. And the rates of those kinds of errors can vary with the underlying rate of infection, the number of tests being processed by a given lab, the level of training those lab workers have, the number of hours they're putting in (tired people make more mistakes), and probably other factors I'm not considering.
