Clavier = French term for keyboard. Clavier = German term used during the Baroque period (and now, as you have noted, also in English) as a generic designation for .any. keyboard instrument. Later, i.e. the era of K.Ph.E. Bach, it denoted mainly the clavichord. During the 19th-C the term identified the piano exclusively (in modern German, 'klavier***'

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Be enlightened; just follow the variables, below -
Quite often overkill is present: WTK I or WTK II, identifying the Volume - WTK being a two-volume work; Number j in Key of K (major) / k (minor); BWV xyz. You will most likely see this overkill presented as Prelude and Fugue No j in k#, BWV xyz, WTK I. You may see a shorter form, such as WTK I/j. No matter, 'tis the BWV xyz which is of the greater import.
You need to consider access to an appropriate reference book - or three!
(Interesting description that -> 'too baroque' -> Great stuff!)
Quite possibly spurious ... both! The performers' sources .&. your score in hand are equally capable of being accurate ... or not.
You are describing, w/some accuracy, an 'inverted mordent'. Introduced into the literature by K.Ph.E. Bach some time after 1750, the manner of indication on the score and interpretation through performance practice has changed somewhat over the years. Do find a reference tome ... W/some degree of certainty, I suggest you'll also encounter the 'pralltriller' and 'schneller'.
One reference which will not harm is K.Ph.E.'s rather famous 'Essay'. You would need more recent sources, of course, to follow the changes in writing and performance practice during subsequent centuries and up to the present day.
*** FYI: LvB quite emphatically indicated that the five late sonatas, not just one in particular, were written for the (hammer)klavier.
HTH.
Best wishes, Scott!