Lots of options
Generally armoured from source if it needs to go underground or in an area at risk, option is to use trucking for that as well and non armoured inside.
Its a bit about look, do you want the neatest (underground cable to wall, sunk cabling to lights etc, or a consistent look of trucking across the whole)
Access of course is also an issue, its why many places tend to have surface trunking as opposed to sunk.
Sunk trunking gives you the best chance of being able to pull a cable in future should something happen (like it being drilled through) if you want the minimalised look
Most people seem to use pond cable for outside wiring, as your only probably going to pull something like 100w its hardly anything, assuming your going LEDs
The double socket however leads me to think your probably better running an armoured cable to that, then spurring off that to a 3 amp circuit for the lights which is how my garden is wired. You need to consider this is max at 13amp of course for the sockets and lights so depends what you planning to run, if its close to that or more you would need a dedicated line put in rather than a spur.
You mention a pump house so thats probably already got a proper spur to it?
Just to link you the pond cable
https://www.screwfix.com/p/time-318...ns7P00puZgXiJEJG7eYaAt_OEALw_wcB&gclsrc=aw.ds