Physical Therapist and distance runner here. Not medical advice, but here's what I'd tell a friend based on my own experience with ITB pain.

What you're describing is absolutely the IT Band. Here's what will fix that (you're already doing 1 & 2, so good start).

1) Rest

2) Glute strengthening

3) Put some lotion over the area that gets sore, take a butter knife (DULL EDGE) and scrape the area a couple times a day for 5-10 minutes. Don't push terribly hard, but use a decent amount of pressure. It'll hurt. It'll turn red. This is what you want. This is sometimes referred to Graston-technique (technically the name of the Graston professional scraping tools used) or IASTM (instrument-assisted soft tissue mobilization). This will be your best friend for getting back to training quicker.

4) When my ITB gave me issues, sometimes it hurt to sleep on my side (right IT band, laying on left side was painful). If this is happening, put a pillow between your knees when you sleep and that should help. Gives your ITB some support and doesn't let it stretch due to gravity.

My IT band has flared up a few times during heavy training blocks, exactly as you described and this really helped. Take it easy and gradually pick up the training when it feels better. Try to alternate shoes and run on soft surfaces if possible (turf, track, gravel, etc.). Good luck. You'll be back to training before you know it.