Apparently you've got some fairly unusual hand watches and stopwatches. This wouldn't normally be true, but it could be in some particular case.
The least count is just the smallest measurement difference we can reliably distinguish using that measurement tool.
In general most stopwatches record data in centiseconds (0.01 seconds), so they would have a least count of centiseconds, unless we account for the fact that it takes a little bit of time to actually push the button, in which case maybe deciseconds. (This is why Olympic events use automatic timers instead of manual stopwatches---because events can be won or lost by deciseconds.)
But in this case, it sounds like we are dealing with some particular stopwatch that only has a digital display in seconds, and therefore there's no way for it to have a least count any smaller than one second.
We can compare this to an analog watch, which in principle could have an extremely tiny least count, if it were somehow made so reliable that it actually lined up each atom in the right place to represent that exact nanosecond. In practice, most analog watches actually click forward second by second, and would have a least count of 1 second; but if it moves smoothly and still keeps proper time, then its least count is the smallest amount of time that we can reliably distinguish by looking at it, which could very well be one fifth of a second, or 0.2 seconds, or even less if it were made extremely precise.