I think it depends on the tool and what its being used for. I can only speak of my experience with the tools I own. Everything I own is Milwaukee M18 Fuel and the batteries range from high demand 9.0Ah to XC 4.0Ah. I don't think the batteries have any effect on the tools performance, just run-time.
The tool that is used non stop most days on the job is the 7-1/4" circular saw. I do notice a slight difference when the battery is low, but not enough to change out batteries before a total discharge. It's hard to say for sure because I think the sharpness of the blade has a lot to do with it.
My 1/2" hammer drill gets used on the job most days, mainly for drilling 3/16" into concrete. I don't notice any change from low batteries with 3/16 bit. Now and then I'll drill a few 1/2" in concrete and there is a huge power loss when batteries are low. I change batteries when 1 or 2 bars show on battery.
My right angle grinder sees quite a bit of use. Used some on the job with a diamond wheel to grind concrete, used more at home for cutting or grinding steel. There is somewhat of a loss of power and a change in sound when the batteries get low. I usually change batteries with 1 bar showing.
Impact driver I use mainly at home, sometimes at work, never noticed any difference with low batteries.
I only had a couple of months last year with my string trimmer at home, so not many chances to go from full charge to totally discharged batteries. Only time I did battery change was when I started with 3 bars showing, but I did notice a power loss and changed out batteries with 1 bar showing.