There are three types of ”cross” shaped screw.
Phillips is the basic cross shaped thing. Designed to cam out. Drivers are often labelled PH0, PH1, etc. Seem to be quite rare in the UK these days, where Pozidriv is much more common. When you think you find one consider the source of the equipment. If Japanese, think about JIS instead.
Pozidriv is sort of cross shaped, but has a subtly eight pointed star, kind of like a compass with NE, SE, SW, And NW being smaller than the four main cross directions. You can see the difference in the screw heads. Drivers are often labelled PZ1, PZ2, etc.
JIS is very like Phillips, but with subtly different dimensions and angles. Screws may be marked with a dot on the head. Less prone to cam out than Phillips. JIS drivers may work acceptably in Phillips screws, but not the other way around. Drivers can be quite hard to find. Vessel make some OK ones. Not sure if there’s standardised driver labelling. Vessel’s are P.1, P.2, etc. Others just #1, #2.
Just as important as getting the right shaped driver, is getting the right size. Just because it sort of loosely fits and let’s you twist, doesn’t mean it’s the right size. A good driver into a new screw is a very satisfying positive slop-free fit, and it’s worth learning what that feels like.
And yes, the limit adjust screws on lots of Shimano derailleurs are JIS. Curiously, for such small screws, they use the #2 size driver rather than the #1. They get chewed up easily with the wrong driver, or, if the derailleur travel is still up against the stop as you try to adjust it.
For hex keys, try Wera Hex-plus.