How to get rid of "white border/matte" on my GIF while retaining some of the transparent parts?

Hi all,

I animated this GIF over After Effects, I then exported it with AE and while it also had the transparent reflection that I had on the phone, the matte or borders were just too fuzzy for my liking, see GIF below:

What I tried to do then was import it to Photoshop and then try to export it as GIF, however I only have these two options right now that I know of:

This is to be uploaded on a web page as a graphic so I would like to learn if there's any way I could retain most of the quality?

Thank you.