[RPG] difference between types of musical instruments and actual musical instruments

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.

Yes, the Entertainer background is exactly that - a 'background' feature. It reflects what the PC did before they took up the life of adventuring. The overlap of having two musical instruments thus applies. It reflects that your Bard character's entire life has been immersed within the world of entertainment - as such, they have mastered (and possess) a wider range of instruments than a person beginning their career as a bard might normally expect to have. Do remember however that the DM is the final arbiter of their campaign and you should always run questions by them before simply accepting anything someone unaffiliated with your group says.