Purchase-money second liens are exactly what you would expect, given their name: they are second liens that are taken out for the purchase of a property. They are usually done as a way to avoid private mortgage insurance (PMI), to avoid having an escrow account, or to put less money down on a home that would normally require a jumbo loan. Second liens always carry with them a higher interest rate than that of a first lien.
The problem with using purchase money second liens as a way to avoid PMI is that, in most cases, their presence now makes the rate on the first lien higher than it otherwise would be. This used to not be the case, but now it is. Also, PMI rates have come down tremendously over the last couple of years. For these reasons, it normally does not make sense get a second lien just to avoid PMI.
The first lien rate increase that comes with the presence of a second lien has been in place for a while, but it was greatly expanded by Fannie Mae and Freddie Mac in 2015. This has caused a dramatic shift away from second liens and toward PMI. We are not sure why Fannie Mae and Freddie Mac instituted this change. Normally, the reason for these types of changes is risk. But we have trouble seeing how that applies here: theoretically, Fannie and Freddie's risk exposure does not change whether there is a second lien or PMI. It is possible that the presence of a second lien causes more problems for the first lien holder in the event of a foreclosure.