In the starting position I've attached, would I be better off moving one east to get the extra commerce/ocean squares or to stay put and getting the extra hammers from hills?
Stay where you are. In FfH2, unless you are Lanun: Water Tiles < Land Tiles < Riverside Tiles < Water Resource Tiles < Land Resource Tiles < Riverside Resource Tiles
The three

shown on those Coast tiles is attractive, but in practice it is hard to access successfully in FfH2. You must research Fishing before you can use them, and then must research Sailing and build a Lighthouse before they can support themselves. Other than the Clams tile, working any of those coastal tiles will hurt your growth and expansion much more than the Commerce will help your research. Unless you are planing to follow the Octopus Overlords religion, researching Fishing in the early game to allow you to work one tile (the Clams) is probably not worthwhile.
I would suggest researching Farming > Calendar > Animal Husbandry as your first three techs. This will give you 6 decent tiles to support growth and settler production. Where you go after that will depend on whether you plan to adopt God King (to leverage the

potential of your starting location) or the more natural Aristocracy (the

bonus from Aristocracy farms is too attractive to pass up as a Financial leader). You have no mineral resources (shown) to benefit from mines, so Mining would probably not be a priority tech and so you won't be producing enough

to make God King worth passing up on Aristograrian Farms. I would suggest Ancient Chants > Education > Code of Laws as your next three techs.
Also, in general is it better to aim for production (hammers) or commerce (gold) starting positions in FFH? Or does it depend on leader/situation?
It depends mostly on your leader, and somewhat on your situation, but it's a question of

vs.

rather than

vs.

. You want to have as much

as possible, without sacrificing your growth or settler production. Growth means food, and food resources are usually the first to be utilized in order to kickstart growth. What happens next depends on what civics you plan to use.
Your leader's traits usually determine what civics you use. For example, Financial begs for Aristocracy whereas Expansive (encourages use of

to produce settlers) and Spiritual (push to found a religion first mean researching Mysticism early anyway) fit in nicely with God King. Early aggression also favors

, so Aggressive leaders may be inclined to use God King to further increase military unit production.
Your situation is also important. If you plan to adopt God King then

become quite valuable in your capital (allowing you to produce early settlers in a timely fashion with

instead of

). Lots of hills and/or mineral resource tiles benefit from God King, and so if you will be using that anyway you may want to place your capital to maximize your

output. With Aristocracy it's all about the farms. Lots of flat Grassland and Flood Plains tiles begs for Aristocracy, because those tiles can also benefit most from Agrarianism (the natural companion civic to Aristocracy). Flat Plains tiles can be used for Cottages or Workshops, but too many of them and there may not be much benefit to using Aristocracy.