Proficiency bonus is noted in your character advancement table (in Character Creation file, or PHB p.15) as "Proficiency Bonus", and that is universal for your character level. For example, according to the table, a 3rd-level character gets +2 for proficiency bonus, because the table says so. For convenience of single-classed characters, you can also look up your proficiency bonus for your level in your class advancement table.
Thus, a 5th-level Fighter with Strength 18 (modifier is +4) wielding his greatsword makes attack roll with 1d20+4 (Strength modifier) + 3 (proficiency bonus at character level 5) = 1d20+7.
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.
Best Answer
There are some trends based on the slight differences in meaning.
The books use the different wordings based on when you would normally use them in the English language.
The official D&D Style Guide (p. 8 in v. 1.08a) addresses the difference between 'in' and 'with':
Looking through all the books there are some clear other trends too:
In conclusion, there are clearly some preferred wordings used depending on the context. There are obviously exceptions, so it's best to always think for yourself which wording works best in your situation.