Why is "hole" conductivity called so, isn't that a case of electron movement?
You are correct that the only thing moving in a metal or semicondcutor is electrons, but also that the behaviour of a hole and an electron is different.  A couple of ways in which this is important for electronic engineering (as opposed to solid state physics) are:


At a PN junction.  In the semiconductor far from the junction, the behaviour is pretty much the same whether there are hole or electrons doing the work. But at the junction they can meet, and when they do, the electron falls into the hole. Or to use the technical term, the electron and hole recombine.
In a hall sensor.  Electrons and holes which are moving are pushed sideways by magnetic fields.  They have opposite charges and are flowing in opposite directions, so they are pushed the same way. But because they have opposite charges they create opposite voltages.  The hall voltage in a P type semiconductor is in the opposite direction to the hall voltage in an n type or a metal.


Also, if you get stuck into the physics of conduction in metals and semiconductors, you quickly find that the quantum nature of electrons mean that you can't really think of individual electrons or holes moving about.  The real physical insights come from something called a Fermi surface. The idea of electron surfaces and hole surfaces is key to those insights, so after learning about them, the "electron" and "hole" terms seem more natural.