The "R" under "Ballot Type" stands for "Regular," not for Republican.
Good catch zeestephen but I think you got it backwards.
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 printout posted in the sloppy blog post* that links back to the Daily Mail article, the registration date is the date the person first registered to vote in Manhattan (and Trump has said he prior to 1987 was registered in Queens), and the Enrollment, i.e. Republican, would indicate the voter's party registration at the time the printout was made.
*Sigh. Bloggers. Some are good but I find a lot are very sloppy in their research. It only took me a few minutes to research the above. Perhaps I should start a blog about bad blogs : )