Hydraulic, solid or roller lifters - they are the interface between the cam and the push rods.
Tappets came about because early hydraulic lifters tended to leak down and cause a "tap" that sounded like solid lifters out of adjustment.
The obvious problem with hydraulics is that you can't adjust out the noise.
Many, many moons ago we used the term tappets quite frequently with customers who were complaining about engine noise, especially with the higher performance cars.
"they wouldn't call them tappets if they didn't tap"
Then we would try and convince them that it the valve train is tapping just a bit at least the valves aren't burning (big problem when leaded gas disappeared).