You should also take note that in different editions, there are different ornaments, even among the "urtext" editions like Henle or Vienna or Koeneman especially in the earlier sonatas. Some sound better than others and some make more sense than others and this has led editors to alter the originally written ornament to one that would be easier to interpret with modern eyes. It may not be what was originally written but it was what was intended to be performed. For example, a written mordent [e.g. CBC] could have been intended to be performed as a turn [e.g. CDCBC] during the time it was written. I think this is called the Haydn turn.
But considering that these ornaments can be used as a key motivic element such as in the A major sonata Hob.XVI:12 first movement, it is clear that the motive can be changed during the repetition. The opening motive that begins this piece was originally written as a mordent [AG#A] but editors have altered it to be a turn [BAG#A] as it probably should be performed (and sound most pleasing). This opening motive occurs at the end of the exposition which leads to a satifying return of the opening material during the repeat. However, after the repeat has neared conclusion, this motivic ornament should probably change to a mordent that leads to the development section which is a series of 3 mordents leading into a trill; the mordent at the end of the exposition satifyingly leads to the developmental material.
This is just one example of written ornament being performed in a different manner, at least viewing it with modern eyes. This is probably due to the fact that Haydn was not so particular in notation considering this sonata was clearly meant to be performed on a harpsichord, and not any other keyboard instrument at the time of composition.
Which leads to the issue of "authenticity". Ornaments were viewed to be rather subjective even in Haydn's time. Not every keyboardist was acquinted with "proper" execution even though we have some treatise like those of CPE Bach and Couperin (and who in those times were able to read them?) Keyboardists performed them using their personal tastes even when an ornament was not written into the score. This is a very important aspect to performance: written notation and performance practice varied widely. For this reason, there is no such thing as "authentic" performance practiced and it is clear that Haydn wasn't very particular especially during his early sonatas though his practice changed to be more specific in his later years.
Topic: ornaments in Haydn