A "cover" would mean simply a version of any song that was popular before you did it, also known as standards--Look back and you'll see that popular songs, or standards, often had several simultaneous or nearly simultaneous hit recordings--

Do you think that Frank Sinatra had a big hit recording of "MY WAY"? His version of it(actually a cover of a reworked French popular song, Comme d'Habitude) was only on the charts for eight weeks in 1969, and never even made it into the top 20--

Elvis actually had a bigger hit with this song than Frank, in 1977--At any rate, it became Frank's trademark because he he featured the song in his act over the years,but it was Elvis' trademark, and also Paul Anka's, who worked up the English version--

Some people may think it's a Sinatra song, but then, I know people who think Willie Nelson wrote "Blue Skies"--