What are the choices for proficiency in a musical instrument?
The rules say the following regarding musical instruments (PH p.154):
Each type of musical instrument requires a separate proficiency.
Since this text is an annotation for the equipment list on the same page, one might assume that a "type of musical instrument" refers to entries from the list. So "flute", "horn" and "pan flute" would all be distinct proficiencies.
However, Pipes of Haunting and Pipes of the Sewers both require that (DMG p.185):
You must be proficient with wind instruments to use these pipes.
This suggests that the much broader class of "wind instruments" is a "type of musical instrument" for the purpose of proficiency.
Bards choose proficiency in three musical instruments. What are their choices?
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.