Strange Behavior

[1FJEF]

Same tire & tube. (An old Specialized Crossroads 26x1.95")
Two different (559mm) front wheels, both bolted axles, one is 26mm external rim width, the other is 32mm.
The tire is fine on the 32mm wide rim, you can see a tiny bit of lateral run out, but nothing you would give a second thought to on a commuter/utility bike that never breaks 20mph.
When mounted on the narrower rim it's a mess. The lateral run out is awful, no way would you ride it. There's even a strange large cupped area at the sidewall/tread junction.
No matter how many times I remount, re-position the tube, fuss with the bead and pop air in the tire fast, I can't get it to behave.
A Schwalbe 26x2.125 tire is fine on this narrow rim. The offending Specialized tire is fine when remounted on the wider rim.
Frustrating.

[nfmisso]

Which Specialized? Is it an Armadillo?

The Armadillos have a very rigid protection layer under the tread, which does not like to be deformed - it will rip the tire apart if it is. Armadillos have a MINIMUM inflation pressure specified on the sidewall; I suspect for this very reason.

[1FJEF]

It's an old discontinued tire. I believe you are correct, the narrow rim has a deformation at the lip which (which I can't see) that is pulling on the more rigid structure of the Specialized & deforming it, while the Schwalbe tire just stretches & accommodates it.
I don't have any more spare bolted front wheels and I need the wider one to go back on another frame. It's a pity, I really like that Specialized tire. It is old, no cracks in the sidewall, marked Crossroads EX. The protection is labeled "Flak Jacket".
The sidewall actually only measure 47mm wide, the tread lugs make it look wider. A narrow, tall, center "zig zag" belt of tread makes the ride responsive, and feels quick while still giving a hint of cushion.
Was that silver front wheel from your Green Bike bolted or QR?