What signal does being prepared to stay overnight on a Saturday give to the airlines? [closed]

What signal will "a trip with an overnight stay on Saturday" give to the airlines and how would they possibly react?

Almost none at all.

The accepted answer used to be correct maybe 15 years ago, but much less so today. Prices are set by complicated revenue optimization systems that apply data mining on booking behavior and patterns, historical data, current booking rates and artificial intelligence. Apparently there isn't a whole lot of actual correlation between "willing to pay more" and "not staying the weekend" and as a result the price difference between Sat night stay or not has pretty much disappeared.

I just checked a a routes I regularly fly and there was almost no difference between a Tue-Thu (same week) or Tue-Tue (including Sat night) trip. It's more expensive on Mon and Fri since more people travel on these days; non-stops are more expensive than connections; but I couldn't find any significant price difference for a Sat night stay.