After the 2019-03-09 Canvas Release, which included the addition of a Commons Favorites Importing icon in the rich content editor (RCE), our Google integration icon "disappeared" from our RCE.    Well....it really didn't disappear, but that's what a lot of teachers thought, and they panicked.    The Google integration moved underneath the "more external tools V icon.
I submitted a ticket (Case #04164555), asking how we could get our Google icon back out, for easy access. I asked if we could re-organize our apps, for example by moving another app icon (CK-12) under the V, and/or even the Commons under the V, to free up space for our Google icon. Teachers use the Google icon a lot so it's important to have easy access and reduce clicks.
I was told "There is not a way to easily change order of the external tool icons that display in the Rich Content Editor.  ....the external tool icons will appear in the order that they were installed into Canvas.  It looks like the CK-12 was the first external app installed as a Editor Button, then the Google Apps was next.  You could delete the CK-12 app which would then put the Google Apps icon in the Rich Content Editor.  If you then re-install the CK-12 app, it would be added to the bottom of the external tools list in the Rich Content Editor."
It would be great if system admin can re-arrange the external tools in a way that makes most sense for our district.

This idea has been developed and deployed to Canvas

For more information, please read through the Canvas Deploy Notes (2020-08-26).