While reading the D&d 5E Basic Rules, I noticed that under "Other Dexterity Checks", it said "Play a stringed instrument". This led me to wonder what ability is used for playing other types of instruments- specifically, woodwind instruments (it seems that woodwinds and strings make up 80-90% instruments listed in the 'Tools' section of Chapter 5). I searched everywhere, but couldn't find anything.
[RPG] What ability is used for playing a woodwind instrument
dnd-5e
Related Solutions
The scope of "types of instrument" is illustrated by the PHB
The full text of the passage you quote from says:
Musical Instrument. Several of the most common types of musical instruments are shown on the table as examples. If you have proficiency with a given musical instrument you can add your proficiency bonus to any ability checks you make to play music with the instrument. A bard can use a musical instrument as a spellcasting focus, as described in chapter 10. Each type of musical instrument requires a separate proficiency.
Here it is clear that 'types of musical instrument' are precisely those given in the table, namely: Bagpipes, Drum, Dulcimer, Flute, Lute, Lyre, Horn, Pan flute, Shawm, Viol.
The list is not meant to be exhaustive, but it does show the scope of 'types of instrument' - each entry is both a "type" and a "given instrument" as per the rule above, and can be used as a basis for selecting other appropriate (types of) instrument.
The magical item descriptions are meant to be inclusive
The magical item entries you give by contrast are meant to be more inclusive than proficiency with one narrow 'type of instrument'. "Wind instruments" as given in the spell descriptions include the 'types of instrument': bagpipes, flute, horn, pan flute, shawm, or any other wind instrument suitable for your setting. So if you have any of those proficiencies, you are good to go. The magical item description is not however meant to be a key to understanding what 'types of instruments' are - this term is illustrated clearly enough in the PHB.
Reference the tool proficiency optional rules from Xanathar's Guide to Everything
You can't apply your proficiency bonus more than once to a single roll, as you've established; it seems like you should probably make this roll using your expertise in performance, since performance is a relevant skill and with expertise that grants you the best modifier. However, there are options to represent your particular combination of proficiencies beyond simply adding more bonuses to the roll.
Xanathar's Guide to Everything, in the Dungeon Master's Tools chapter, includes a section titled Tools and Skills Together, which addresses exactly this situation where a character might have overlapping tool and skill proficiencies. It makes a couple of suggestions about how a DM could adjudicate the situation:
Advantage. If the use of a tool and the use of a skill both apply to a check, and a character is proficient with the tool and the skill, consider allowing the character to make the check with advantage.
[...]
Added Benefit. In addition, consider giving characters who have both a relevant skill and a relevant tool proficiency an added benefit on a successful check. This benefit might be in the form of more detailed information or could simulate the effect of a different sort of successful check.
The chapter then goes on to describe the possible benefits of various tool proficiencies, and gaining advantage on a performance check incorporating that instrument is given as a benefit of proficiency with a musical instrument:
Skills. Every tool potentially provides advantage on a check when used in conjunction with certain skills, provided a character is proficient with the tool and the skill.
[...]
Performance. Your ability to put on a good show is improved when you incorporate an instrument into your act.
An extra benefit to a successful check might perhaps be allowing the bard an automatic success (or at least granting advantage) on a subsequent social skill check to influence the audience after the performance.
The intent of these optional rules from Xanathar's is to make tool proficiencies more valuable, in order to encourage players to take and make use of them as character options, and reward characters who have invested in such specialisation.
Best Answer
The things listed under each ability are just examples, not a complete and exhaustive list of the only things possible with ability checks. They're listed to give an idea of what generally falls under each ability, and provide exemplars that DMs can extrapolate from to unlisted situations encountered during play. Based on the things listed under each ability, playing a woodwind—or any kind of instrument—is also a Dexterity check.
I could see an exception for an instrument that required no manual dexterity to play right but did require massive lung capacity, and ruling that a "performance" with that instrument was a Con check. But such unusual circumstances are precisely why the abilities are described using examples, and the exact check required is then left to the DM to decide based on that guidance and their own judgement.