Hi Manolo,
Take a look at this article:
http://www.pihakaetta.com/sinhala_weapons_armor_low.pdf
Page 8 mentions different names for piha variations. I've noted this before, but "piha kaetta" is a Stone term that is a little contradictory in Sinhalese, but it is in fact used to describe one specific type of chopping knife. "Piha" denotes the decorative knife like we are discussing. "Da Kaetta" is a more utilitarian chopping knife.
Anyway, to answer your question, yes there are many specific names for piha variations such as Ul Piha (the slender straight example you posted), Gal Mita Piha (the kind with the crystal or jade handles), Vak Piha (curved blade), etc.
Sorry, the PDf is a rough scan, but readable if you're interested.
-d