My home insurance policy says this (my bold)
"We will pay all amounts you, or any member of your family, become legally liable for as a result of living in your home, or for any other reason, as a result of:
a) accidental death of or bodily injury to any person; or
b) accidental loss of or damage to property;
caused during the period of insurance shown in the schedule.
But not liability for
i) death of or bodily injury to any member of your family or
your domestic staff;
ii) damage to property belonging to or in the custody or control of
you, your family or domestic staff; or
iii) claims arising from the following:
[deletions]
4) You or any member of your family owning, keeping or using any
a) motorised vehicles
b) caravan while being towed;
c) watercraft, hovercraft or aircraft other than hand-propelled
craft or models; or
[and so on]"
I've taken that to be third party liability cover against basically anything that doesn't involve a vehicle or a power boat. It specifically (to me) doesn't exclude using a bike.
I've CTC insurance too, which feels pretty similar, but has cycle-specific legal cover.