No, sensitivity is TPs / all positives (detected TP and FN). It’s the green half of the diagram in your link. The language is very easy to get tripped up by though. :)
For me it’s easier to think of sensitivity as being “sensitive to the true positive” without saying much about false positive or negative.
A sensitive test will catch many positive cases. It may or may not have false positives though, eg a test that’s always gives the right answer vs a test that always returns positive no matter what.
A specific test will give you few positive results when the true answer is negative. You could use it to rule something out. One test might say “patient has A or B condition”, and a second test with high specificity may then rule out A or B, leaving B or A, respectively, as the probable condition.