1) I think what you are seeing are the child texture controls being displayed, not the "up" & "over" textures at the same time.
For example, mailButton has 2 children (texture controls) that also get displayed in certain states, like the glow gets displayed when you have mail..no clue what the burst is for. For example you have:
Lua Code:
ZO_ChatWindowMail (which is: CHAT_SYSTEM.mailButton)
-- Which has two child texture controls
ZO_ChatWindowMailBurst
ZO_ChatWindowMailGlow
The glow is not an "over" texture. The "over" texture just makes the mail button texture a little brighter. The "glow" texture is a separate texture that outlines the button texture with a glow.
The other controls your working on have the same thing:
Lua Code:
ZO_ChatWindowNotifications (which is: CHAT_SYSTEM.notificationsButton)
-- has 3 child texture controls
ZO_ChatWindowNotificationsEcho
ZO_ChatWindowNotificationsGlow
ZO_ChatWindowNotificationsBurst
The friends texture doesn't have any child texture controls.
If you don't want the glow just get rid of the glow texture on the glow texture control.
2) "So, in this case, I also need to see if it is possible to only show my "glow" version when there is an unread notification, instead of the "up" version and the glowing animation that is also showing."
Um, not without a lot of work...I don't think. The problem is the "up" texture is used when you have mail & when you don't. You can't just get rid of the "up" texture because then when they don't have mail there will be no button. You'd have to write code to figure out when the have mail & when they dont & change the texture accordingly.
Honestly I think it would be a lot easier to change your textures to match the way the game does it.
Create your textures for the: Up, pressed, over
that the user will see when they have No mail (or notification).
Then create any glow, echo, burst effect textures that will be displayed WITH the main textures (for when they have mail & a notification) and set the Glow, Echo, Burst child texture controls to have those textures. If you don't want a glow you could just get rid of it or make it a transparent (blank) texture.