In this case the third party could be the employer that pays the salary, or the bank that receives it.
The decision to trust the issuer or not is down to the entity viewing the information, and depends on the particular requirements of the situation. A zero-knowledge proof that I am over 25 issued by my local library may be enough for the local theatre to let me in to watch a suitably rated movie, but not for the local bar to let me purchase liquor (in the latter case a proof directly from a government entity would be more appropriate).
Because it's written by someone who (as they admit) first learned about it recently from a tweet. It appeals to laymen very well, but conflates "Zero knowledge" and "Zero knowledge proofs" throughout the article. Not sure if that's intentional or not.
Edit: Learned it was important from a recent tweet*. it seems the author may have already known what they are.
Monero is referred to as a zkp coin in this article; but afaik it is cryptonote based, using a combination of mixing and stealth-addresses and doesn't actually use zero-knowledge proofs. Did that change at some point or was this said in error?
I think this is still a fantastic article as an introduction to the concept, I hope it is not removed.