No. It wasn't debunked. Some blogger thought he or she had but was very sloppy in their research, assuming that the R in the column "voter type" stood for Republican. It doesn't. Furthermore, if Trump had always been a registered Republican, I would have expected him to come out with a denial to those saying he's changed party registration several times. Crickets.
See page 32 of this 2013 Annual Report from the Board of Elections - The City of New York:
http://vote.nyc.ny.us/downloads/pdf/documents/boe/AnnualReports/BOEAnnualReport13.pdf
In the column Voter Type, the R stands for Regular Voter as opposed to: E-Special; F-Federal; H-R(Mail Address); M-Military; O-Overseas or P-Permanent Absentee. I am not sure what all those designations mean but none of them have to do with party registration. Military, Overseas and Permanent Absentee are fairly obvious, but I can't find any keys or designations for the others. But it has nothing to do with party registration at the time the vote was cast.
Ballot Type would be R for Regular or A for Absentee.
On the voting record print out, the registration date is the date the person first registered to vote in Manhattan (and Trump has said that prior to 1987 he was registered in Queens), and the Enrollment, i.e. Republican, would indicate the voter's party registration at the time the printout was made.
