I used to get really nervous when playing in front of people. I found my footing with two things; playing consistently and being so practiced that I didn't have to focus on my guitar at all and just had fun. Thankfully I was introduced to Dimebag/Pantera via the home videos really early in my band days and I'm more influenced by that guy's spirit/fun loving attitude than his playing. Whether I'd be messing with my friends onstage and trying to screw them up or just enjoying the music we were playing (lots of Metallica back then), those were my happy places.
For years I didn't think I'd be able to get into the music onstage unless it were metal, but in the cover band I'm in now I'm having just as much fun playing Harry Styles songs as I ever did anything else. I was especially that way after writing original music and gigged my first original band for several years. When that band imploded I wasn't sure if I'd ever have that much fun onstage again and while it hasn't been as fun consistently as it was with that band (we used to drink a LOT and our shows were just big parties with our friends and co-workers) I still have a great time, even without the booze involved.
Also, I've bombed onstage so many times, on vocals and guitar, there's really nothing left for me to be concerned about, I already know what that's like.
I never dealt with Red Light Syndrome, the studio has always been my favorite place to be, but I think it goes back to being so rehearsed that I knew I wouldn't fuck up and instead of paying attention to everything I'm playing, I go on autopilot and just focus on the click. For 6 years I thought I'd be content never playing live again because I was enjoying the studio so much, but eventually I got the itch to play guitar live again.
One thing I used to do when I was purposefully trying to be more interactive with the audience/make eye contact was just staring straight through the audience. Whether or not someone was in my line of sight wasn't the point, because I'd have my attention behind them or to the sides, but it looked like I was looking right at them.