There are others on here more knowledgeable than I, but as I understand it:
You can only have 1 "master" socket per telephone line, although you can have other sockets on the line. It's better to "daisy-chain" them rather than use "star" wiring topology.
You can only have one modem per telephone line, and it's this which sorts out the broadband signal and passes it to a router for distribution around the local area network (LAN). These days modem and router are often combined into one unit, which is technically a modem/router, but often referred to as a "router". The modem doesn't have to be on the master socket (although it's usually better if it is.
In short, you can probably move the master socket into the house, run the router from that, and keep an extension telephone line to the summer house. You will need to run a separate line from a LAN port on the router (or use WiFi) to the summer house.
Of course, the Openreach engineer will be able to see the actual site configuration, so will be in the best position to advise.