LAN<-->WAN<-->LAN ; L3 switches or Routers??

We have a case, where there is a LAN in the local place and a separate
LAN in the remote place. This also means, there is a WAN in between
these 2 LANs and they communicate via this WAN.
I would like to know
- What is most preferred among 'Routers' and 'L3 switches', which are
to be used for routing the messages from the LAN to the other LAN, via
the WAN. Could anybody give a rationale behind that choice?

Also, I would like to know whether this choice depends on
- whether the WAN is our own or 3rd party network?
- whether there is only one VLAN, for which some of the machines are
kept locally and the other ones remotely

It is quite common to use layer 3 switches connected directly to a
carrier provided WAN transport as many carrier deliver the WAN via
Ethernet..