If I'm not mistaken, it's a completely different product with the same name. It was lync for a long time, and they just basically changed the name but kept all the innards intact. I'm fairly sure it's on prem.
Yup, completely different products. Skype for Business is pretty much just a rebranding of Lync 2013 - for the longest time, the Skype for Business client was still calling itself Lync 2013 on it's about page. It's been pretty confusing, to be honest, since the two different Skypes are completely different products, talking different protocols that just barely communicate with each other.
Skype through Office 365 is also Skype for Business/Lync.
With the Lync => Skype for Business rebranding, some of the original MSN-emoticons that used to work in Lync / Office Communicator have been removed. :-(