OK, so it seems you have two displays that both look exactly as you illustrate when reliably connected to a test rig in which "normal" 1602 modules display perfectly.
That given, there is no obvious answer. What you illustrate suggests either a fault - which is rather unlikely to be identically present in both displays, or a peculiarity of design requiring a quite different initialisation sequence.
Frankly, that is about all that can be said.