I can promise you that 6 months later, this issue is still not resolved. The WindwosPhone forums are filled with this complaint and all they do is claim it's your fault for not putting the headphones in properly or for buying the wrong headphones. Shameful.
I purchased an inexpensive pair of headphones for $4.50, and they work flawlessly. It does take a bit of pressure to put them all the way in the first few times, but then it becomes much easier.
I did notice they person you quoted stated the headphones they have worked on the iPhone. Generally speaking, headphones made for an iPhone, to my understanding, do not work because they are wired differently, as iPhone uses a different standard. Buying headphones made for Android phones should work, however.
I bought an inexpensive pair, and they work great on not only my Lumia 635, but also on my laptop, which has a single plug for both mic and headphones. Both the mic and earpieces work with my laptop, so I suppose the wiring must be the same on my Windows 8.1 laptop and my Windows Phone 8.1 phone, because I can swap the same wired headset between both devices without issue.
If a person has made sure they purchased the proper headphones (not made for iPhone), and if they are positive they have been inserted fully, then it is possible the phone's jack may be defective. Unlikely, to say the least, but it is possible. I would recommend taking the handset to the carrier's store, and testing it in the store. I know my local AT&T store would be willing to assist me with this, but I can't speak for the customer service levels of other carriers. If you're trying to work with Consumer Cellular or Sprint, then best of luck to you, because the customer service isn't very good with those companies from my experience.