"Piano Sonata", for example, can cover the work of Scarlatti
Indeed it cannot. Scarlatti wrote no works for the piano whatsoever.
The majority of his output was written for the harpsichord - a plucked-string instrument which in Scarlatti's time would usually have had two manuals, and most often a number of different 'stops', such as a buff, or options to use either four-foot pitch, or (lawks-a-mercy) a sixteen-foot sound alongside the conventional eight-foot. No dynamic effects are possible "by touch" on the harpsichord - yet no-one berates this same feature on the organ, which is set up in similar fashion. Scarlatti specified the organ in four of his keyboard sonatas.
Some have suggested that Scarlatti might have himself played the new-fangled
fortepiano on occasion. However, his works do not make use of dynamic markings that suggest so. Where he does suggest possible 'echo' effects, these could probably be feasibly (or even perhaps better) played on a different manual of the harpsichord. It would probably not be wrong to play his later keyboard sonatas on a fortepiano, although even the composer could not have hoped that one would always be available.
The modern joanna is an 88-key instrument which features one manual only - along with pedals for partly silencing its din when thought appropriate, or prolonging its clatter when the performer hasn't learnt the music very well. Instead of plucking the strings, it
bashes them from below with hammers, and thus is a
percussion instrument, and no relative of the harpsichord's at all.
Similarly, JS Bach wrote no "piano sonatas", and certainly no "piano concertos" ::) Suggesting that he did perfectly illustrates the paucity of thinking behind retrospectively slapping genre labels onto works, in defiance of all respect for their composers' intentions. It also suggests a bogus equivalence between compositional approaches of, say, Scarlatti and Tchaikovsky from which neither of these two gentlemen stands to gain.