Replacing abnormal outdoor dryer vent

The louvers on my original (20 year old) outdoor dryer vent have practically seized up and I’m in need of a replacement. When I take to places like Lowes or Amazon, I only find the style which requires being screwed to the house. The style I have today is different though. It not only has retention on the circular portion, but also mounts securely to the square housing.

I’m having difficulties identifying what style of vent this is, or if it’s even feasible for me to replace it as-is. Does anyone know what this is called? Should I move to the screw-in style? If the screw-in style, is there anything in particular I should remain cognizant of? Such as sealing the screw holes or caulking the vent?

Thanks in advance.

Also, sorry for poor photo quality. It’s dark, cold, and I’m not ready to clean yet.