FWIW: I googled the following:
phillips-head screwdriver: 1.4M results
cross-head screwdriver: 725k results
cross-head screw: 2.7M results
phillips-head screw: 868k results
Based on the internet usage, sounds like cross-head screw and phillips-head screw driver are the common terms. Seems like a confusing way to refer to two things that you'd use together
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FWIW2
homedepot.com, lowes.com and acehardware.com have no cross-head screws or cross-head screwdrivers. They do have phillips-head screws and screwdrivers though...