For STI shifters (which combine both brake lever and shifter, we need to know if it is for a road bike (short pull) or a long pull.
Separate: this model is only a shifter, the brake lever is separate
Long Pull: mechanical disk, v-brake
Short Pull: caliper, cantilever
If the shifter is separate from the brake lever, put "Separate". Otherwise what is important is cable pull. In general all rim brakes require the same cable pull. V-brakes and cable disk are "long pull". So integrated shifters like Shimano 105 ST-5700 just specify "Rim Brake". As far as I know, there is no such thing as a direct cross pull brake with drop bars. However there are hydraulic.
HLC for the Altus ST-EF500 specifies "V-brake". So perhaps that is the correct term for cross pull. They take a different cable pull, so a brake designed for cantilever won't work directly for a v-brake.
Are there hydraulic combo shifters? Yes Shimano 105 ST-R7020 Hydraulic STI Shifter and Disk Brake Caliper (so that includes the brake