To repeat, I have the same latest versions as you of all 3 Edges. The Edge Bar is with GA, BUT in 182 it will not personalise, attempts to make changes are to no avail.
It WILL personalise in Win11.
Yes, didn't catch all the differences, for example my Search Only Layout is square, not rounded.
Hang on... I just clicked on the Vertical Edge Bar and the controls switched to the right & now the collapse button is there. Nope, this freaky... it's back to the way it was.
And my Sys Tray icon is the same as you show.
https://learn-attachment.microsoft.com/api/attachments/a8c37ff3-b920-4ae2-a110-fd4a51b690f6?platform=QnA with 'puters 🤣
Plus, I believe the shape/contours of the Search Only Layout is likely following my RoundedTB settings.
PS: This is Edge GA in Win11
I just added a comment on my last post, the collapse button only shows when the controls are on the right side 🤣🤣🤣
If I move it the button disappears.
As for the shape of the search slot, it has nothing to do with the taskbar.
I got both the new and the old type on the same Windows 11 machine, just a different version of Edge.

