Should I refer to the following as a "handrail," "guard rail," or a "wooden handrail?"

Have a look at the defininiton of handrails from a few sources:

OxfordOnline :
a rail fixed to posts or a wall for people to hold on to for support.

Reference.com:
a rail serving as a support or guard at the side of a stairway, platform, etc.

Merriam-Webster:
a narrow rail for grasping with the hand as a support.

So, it be wrong to assume that rails are metallic. I agree the word rings in the metallic character as we (at least I do) quickly think of the rail-roads. :)
And as @Edwin Ashworth tells us, we can use balustrade for rails supported by pillars or columns. Here is a good definition about it:

balustrade:
a railing supported by balusters, especially one forming an ornamental parapet to a balcony, bridge, or terrace.

Stair guide rails on houses are good examples of wooden handrails. Anyways, I would just like to add that you can say 'wooden handrail' if you feel that it would add to your explanation or add to the sort of visual or tactile imagery that you are trying to project. Balustrade would be acceptable too but only if you don't mind your audience not knowing what they are initially.