"Covers" (old songs recorded by new artists) are not new unless the arrangement is substantially different or it fits a different genre (which would make it non-Standards, right?).
Now you're treading on uncertain terrain. You make a good point about the difference between covers and remakes. Using the original definitions of those terms, "covers" were faithful copies, usually something along the lines of Pat Boone doing a vanilla version of a Little Richard song. "Remakes" were total re-works of songs, such as the Vanilla Fudge version of Holland-Dozier-Holland's
You Keep Me Hangin' On, originally by the Supremes. A song which, incidentally, wasn't all that vanilla.
Then there are such remakes as the original
Layla by Derek and the Dominos and Eric Clapton's unplugged version, which though an excellent song, was almost unrecognizable compared to the original.
I would suggest looking to the world of lounge music "standards" for guidance. Standards were what guys in tuxedos would sing in smoky saloons when people like Frank Sinatra and Tony Bennett were on top, guys like John Davidson and Jack Jones were in the middle, and guys Bill Murray made fun of on SNL were at the bottom. In the clubs where the bottom tier guys performed, the "house band" was there every week, and knew all the standard songs. That's why they were called "standards". Each week, the new guy in a tuxedo came in and sang the "standards".
There were rock "standards" even back in the garage band days of the 60s. Every garage band was expected to know certain universal rock songs, like
Gloria,
House of the Rising Sun, or
Ina-Gadda-Da-Vida.
Today, in searching for new MP3's, I've stumbled on some modern "standards" that seem to be included in many performers' live acts, even though they aren't recorded hits. Examples are
Lilac Wine, Leonard Cohen's
Hallelujah and
Bird on a Wire. Every big-voiced female singer, from Celine Dion on down, is obligated to sing
The Power of Love and
How am I Supposed to Live Without You.