I have heard a lot of clatter from different sources. I've heard that the X is a slang term from the 1960s in the United States but I have also heard that the X derives from the Greek letter "chi" and was used in antiquity.
Your second explanation is the correct one. Chi is the first letter of the word, "Χριστός," or, "Christos," which means, "anointed." It's just replacing the whole word with the first initial of the root word and adding the suffix.
hi! the study of word origins is etymology. Most good dictionaries will include a short etymological description for words; typically the bigger the dictionary, the more detail they include. There are also specialist etymology dictionaries, like the Online Etymology Dictionary, "etymonline". Here's their entry for Xmas
Xmas (n.) "Christmas," 1551, X'temmas, wherein the X is an abbreviation for Christ in Christmas, English letter X being identical in form (but not sound signification) to Greek chi, the first letter of Greek Christos "Christ" (see Christ). The earlier way to abbreviate the word in English was Xp- or Xr- (corresponding to the "Chr-" in Greek Χριστος), and the form Xres mæsse for "Christmas" appears in the "Anglo-Saxon Chronicle" (c.1100).