"Disc" would suggest a CD. "Vinyl" is fine, as is "record". You can say "album" (which implies a larger record and not a single) or even "LP", which stands for long play.

When vinyl LPs were common (and before), "record" was by far the most common word, but "disk" (or "disc") was very common. (After all, people who played records on the radio were called "disk jockeys.") By the way, in the United States, the spellings "disk" and "disc" are both used, but "disk" is more common. I think that nowadays "vinyl" refers to the medium in general (originally an adjective, now an uncountable noun), and "record" or "vinyl record" is used to refer to the individual item. Because CD stands for "compact disc," the word "disk" is ambiguous; but CDs are never called "records," so "record" is unambiguous. The word "jockey" (in the phrase "disk jockey") actually referred to the practice of turning the turntable back and forth by hand to find the exact starting point of the music, to "cue up" the record. A rarer word was "platter." It was a slangy and somewhat affectionate phrase, used only for popular music. During the age when there was coexistence between breakable shellac 78 rpm records, 45 rpm singles, and 33 rpm long-playing records, it was common to call long-playing records "LP's" or "albums." (The word "album," incidentally, comes from the days of shellac 78s, when symphonies, or other long works, were sold as sets of four or more records, literally packaged in an album of tough sleeves, holding the records, that could be turned like pages.

Not exactly what you are looking for, but in UK we say record, vinyl , or just vinyl