I have solubility s given in g/mL. I understand this as mass of solute m divided by volume of solvent V.
Your confusion is arising from mixing the concept of solubility and concentration. Unfortunately, both are expressed the same way. You have to see the context. Most authors (like good scientists) try to write their experiment explicitly so that others can understand and repeat their experiment.
See the examples in this link: Solubility Concepts
Let us say, we find from the literature that the solubility of NaCl in water is 360 g / 1L. It means that if we take 1 L of pure water, we should be able to dissolve 360 g of salt in it. Note by adding 360 g of NaCl, the volume of water may not be exactly what 1 liter anymore. I cannot calculate concentration out of this information.
Now some authors may refer to concentration when they talk about solubility. For very dilute solutions it does not make a difference. So it is always good to see the context.
I read online I should pour the solute into a measuring cylinder/cup first, and then add solvent until the volume fits the calculated value (V=m/s), but this seems to be at odds with the definition of solubility.
Don't trust the webpages made by amateurs or pages from non-reliable sources. With time you will develop the sense which sources are more reliable. There is a saying that paper never refuses ink. It means one can write whatever on paper, and the paper will never say no, even if it is wrong. The same is true for the web. If we search the web for corona virus, you will find a lot of non-sense. Only reliable sources are to be trusted.