Nowhere in the rules is the dulcimer listed in the device table defined or even described. It may even refer to both types of devices. Since it is not defined in the rules, the table must be defined. Talk to your DM and agree with what you think is best.
Settings and taste, this is also very flexible. Each table decides for itself where it wants to play and all kinds of flourishes that are never in the tradition. Choose something that you think is the most fun and most suitable for the game your table is playing.
The type will almost certainly not play a role mechanically
You mention the action economy as a concern, but there are really no cases in which the bard in battle plays an instrument that is important for the action economy.
No bardic skills require playing a musical instrument at all times. And importantly, a bard does not have to play the instrument to use it as a magic focus that you just have to hold.
The rules do not even define the action economics of an instrument. Normally, the bard uses his actions during the battle to cast spells or other helpful combat activities.
I played a Bard in a campaign for almost 3 years and never had to worry about action action in combat because she never did.
If you want to do this during the fight, select the version of the instrument that you think is the best / funniest one under these circumstances, and work with your DM to make sure they agree.
It's a bad idea to focus on houseruling
They say in a comment:
perhaps a house rule that playing is required as a focus
Do not do that. If you are not the DM, ask them not to implement it because it is a great nerf for bards. If, as you implied, playing an instrument is a sort of action, playing an instrument would actively prevent the bard from casting a spell with that type of action (or delaying it until the next round). This destroys a bard's action-economics and creates a more demanding requirement to cast a spell than any other class by far.
It sounds like the DM is trying to put the rules in the foreground. That's not a good idea. Especially not with the magic rules, because they are already picky enough.