Pine shavings are a popular coop bedding, shavings because they're virtually dust-free and chickens have sensitive respiratory tracts, pine because it is said to deter pests like mites. Unlike hay or straw, it's not quick to mold if it gets wet.
Outdoors, these things aren't a factor. The open air means dust isn't a problem unless you start emptying your vacuum cleaner in the run. The outdoor conditions and minimal hygiene (changing the drinking water as necessary, etc) take care of pests or mold. Outdoor, you can really use anything you want as long as you don't let your run be turned into a mudpool, because that's where pathogens can thrive and infect your chickens.
Hay, straw (although perhaps still don't use these in isolation in wet climates) wood shavings, wood chips, grass clippings, leaves, etc, it's all feasible. I don't think there's really a 'best' option, just see what's available and nearby where you live.
Re Deep litter: You need to sometimes clean out the coop when you use DLM in there because chicken manure holds a lot of nitrogen. When the bedding becomes very deep and hot at the bottom, all that nitrogen can turn into high concentrations of ammonia in the coop's stagnant air, which burns your chickens' airways.
Outdoor you have virtually perfect ventilation so it's not necessary to clean out the run if you do it right. Ultimately it's just one big compost heap, and we all know chickens do just fine on top of those. You can just clear out the bedding whenever you need some compost