#It is whatever type your DM and you agree it is
It is whatever type your DM and you agree it is
Nowhere in the rules does it define or even describe the dulcimer listed in the equipment table. It could even refer to both types of the instrument. Since it is not defined in the rules, it is up to the table to define. Talk with your DM and agree on what you think is best.
Settings and flavor-wise this is also very flexible. Each table decides for themselves the setting they want to play in and all kinds of varieties of flourishes that are never included in the lore or anything. Choose something that you think will be most fun and fitting for the game your table is playing.
###The type will almost certainly not matter mechanically
The type will almost certainly not matter mechanically
You mention action economy as a concern, however there are really no cases where the bard is going to be playing an instrument in battle such that the action economy would matter.
No bardic abilities require playing a musical instrument at any time. And, importantly, a bard does not have to play the instrument to use it as a spellcasting focus you only need to hold it.
The rules don't even define the action economy of playing an instrument. Generally during battle, the bard will be using their actions to cast spells or other helpful combat activities.
I played a bard for almost 3 years in a campaign and never once had to worry about action economy of playing in battle because she never did it.
If this is something you do intend to do during battle then choose whichever version of the instrument you think would be best/most fun in that circumstance and work with your DM to make sure they agree to it.
###Houseruling focus to require playing is a bad idea
Houseruling focus to require playing is a bad idea
You say in a comment:
potentially given a house-rule that playing is required for use as a focus
Don't do this. If you aren't the DM, beg them not to implement this because it a huge nerf for bards. If, as you imply, playing an instrument is some kind of action, then playing an instrument would actively prevent a bard from casting a spell using that action type (or delay it until the next turn). This kills a bard's action economy and creates a more strenuous requirement for casting a spell than any other class by far.
This sounds like the DM may be trying to force flavor into the rules. This is not a good idea. Especially not with the spellcasting rules because they are already finicky enough.
If a DM tried to make this a rule at a table I was playing at, I would seriously just choose a new class.
The way I would recommend doing this if the DM really wants the musical flavor is to allow the bard to play the instrument as part of the action used to cast the spell. That gives the flavor without mechanical issues.