No, it is not a fact that runners should always give way to walkers 
In the real world it's much more nuanced. Things like traffic, sun direction, incline, the surrounding area play a part. If a person is travelling uphill with the sun in their eyes, it maybe much harder for them to see an oncoming person and plan to avoid the other person. We all have a responsibility to each other, regardless of speed.
Generally I make space for other people whether I walk or run. I run solo, I generally keep left, I look around me, I move as appropriate in advance.
There was an incident last year on a narrow, single pavement next to a busy 40mph road. I was running up the steep hill and the traffic approaches from behind. Two young women were walking down side by side. I decided that rather than playing chicken with fast moving traffic, I would keep going on my course. She finally pulled in behind her mate inches from collision.
(I am not using that route at present for obvious reasons, and although the traffic is lighter, it's still the direct route to the nearest supermarkets, and frequently used by emergency vehicles, and I do not fancy having to move a metre into that road for self preservation reasons)
Solo people should mutually make way for each other. Groups should fall into single file to allow others to pass safely.
Running on long grass/ uneven verges carries a higher risk of twisting/ spraining an ankle as it's not always possible to see trip hazards of dips, humps or debris.
Running on the carriageway of a road without a pavement should be on the right to see oncoming traffic, except when it is safer to be on the outside of a bend or approaching a blind summit when you are giving more visibility to the vehicle behind you to react rather than suprising an oncoming vehicle. Exactly the same applies to pedestrians. While you may need to step on to a verge to assist a vehicle to pass safely, it is safer to be on the carriageway than the typical rough verge.