Those songs are not considered covers, since they're not a cover from another artist, but a re-edition of a song that they released before instead.
(Songs in alphabetical order)
1) All You Zombies by The Hooters (old version 1983 / new version 1985)
I prefer the '85 version. The '83 version sounds very unfinished. The '85 version is also the most known one as well. Apart from All You Zombies, the songs Hanging On A Heartbeat and Blood From A Stone from the album Amore, were also remixed appeared on the album Nervous Night.
2) Bringin’ On The Heartbreak by Def Leppard (old version 1981 / new version 1984)
Both are good, but I would choose the original one. Anyway it's my favorite Def Leppard's Power Ballad.
3) Broken Heart by White Lion (old version 1985 / new version 1991)
I would choose the original one because I like the album Fight To Survive more.
4) Cherry Pie by Warrant (old version 1990 / new version 1999)
Obviously the original one. A Glam Metal classic (and also a stripper anthem).
5) Cold As Ice by Foreigner (old version 1977 / new version 1985)
I have never listened to the '85 version which was an exclusive UK remix released as a single. But if it sounds like Foreigner during the Agent Provocateur era, it must be cool.
6) Headed For A Heartbreak by Winger (old version 1988 / new version 1991)
There's no significant difference between those two versions from what I've noticed personally.
7) Here I Go Again by Whitesnake (old version 1982 / new version 1987)
As a Glam Metal fan, I would choose the 1987 version.
8. Home Sweet Home by Motley Crue (old version 1985 / new version 1991)
'85 version is better, but actually Motley Crue were lucky that they made it in the charts in 1991 when Grunge took over.
9) I Wanna Rock by Twisted Sister (old version 1984 / new version 2004)
Actually this is just one of the songs remixed for their 2004 album Still Hungry, which was basically a remixed album based on their 1984 album Stay Hungry. Anyway, I prefer the original since it's the version we all know and love.
10) Missing You by John Waite (old version 1984 / new version 2006)
One of the most memorable ballads from the 80s. The 2006 version which is a duet with Alison Krauss sounds pretty tacky in my opinion though.
11) Open Your Heart by Europe (old version 1984 / new version 1988)
I prefer the newer version here. It's fuller than the older one and it also had a very nice music-video.
12) Rip And Tear by LA Guns (old version 1989 / new version 2000)
One of the songs that were remixed for the 2000 album Cocked & Reloaded. Obviously it was an album based on their 1989 album Cocked & Loaded. Anyway original is better of course.
13) Slick Black Cadillac by Quiet Riot (old version 1978 / new version 1983)
There was a time in the 70s, when Quiet Riot were releasing albums, exclusively for the Japanese market, before they exploded in America with their tremendous Metal Health in 1983, and for many fans the album that paved the way for the Glam Metal genre, along with Def Leppard's Pyromania from the same year. Obviously the Metal Health version is better.
14) Still Loving You by Scorpions (old version 1984 / new version 1992)
Their 1992 Still Loving You album was a Remix album, and the title-track was one of the songs that were remixed for this album along with many others. Still, the original version from the Love At First Sting album is sounding better.
15) Wanted Dead Or Alive by Bon Jovi (old version 1986 / new version 2003)
Of course the Slippery When Wet version, but I believe this is one of their songs that sounds much better when they perform it live.