Don't think I've seen those terminals before, but per the picture, sure can see why they wouldn't fit the socket correctly
They almost look like a fuse-holder terminal (for ATO/ATC fuses)
I meant, where did you source the 4-way connectors? One of the links I provided or did you find locally?
Here are different relay types
In the first three pics, the one on the left is a mini relay
The one in the middle is a micro relay and the one on the right is an ISO280 relay
All are adequately rated for the dual lamps current (min 20A)
Bare Relay
Relay in socket
The ISO280 Relay in its weatherproof socket with cover.
Those ISO relays are super-compact!
The socket is actually not intended to hold two relays - I found that some subtle filing/sanding of the side faces of the two adjacent relays would enable them to fit in a single socket/enclosure.
The socket actually uses sealed terminals and of course you can see the seal on the cover system.
Here is another sealed solution; ends up slightly bigger than a standard mini-relay/socket