I'd like to add to Fred's answer, which is correct. Some more points:
If I correctly understand the $\ce{SiO_2}$ contained in the rock as a
part of a mineral other than pure $\ce{SiO_2}$, for ex. the
$\ce{Si_2O_4}$ part of anorthite $\ce{CaAl_2Si_2O_8}$, is calculated
into the quantity of total silica.
This is not correct. When calculating that acidic or basic character of a rock, you have to take into account the entire rock, in what is known as a "whole rock" or "bulk rock" composition. You don't separate $\ce{SiO_2}$ of various minerals in the calculation.
What I cannot understand at all is whether mass or volume is taken into account.
As a general rule of thumb, rock and mineral chemistries are usually calculated using mass (commonly known as weight in the literature) or molar proportions, and rarely if ever as volume.
Also, I have never seen the per/meso/hypo silicic terms used in the professional literature (or any other literature for that matter). Apart from that, the term acidic originated long long time ago when it was thought that silica was present in the rocks as silicic acid. This is now known to be false, and in the past decades the terms acidic and basic are falling out of use. Silicic and mafic are more common nowadays. Note that obviously, this only works for igneous or metamorphic silicate rich rocks. You would not use that to describe rocks that are very poor in that stuff such as carbonatites, or sedimentary rocks such as phosphorites or evaporites.