I did look at the item you purchased, and at the one I recommended; I actually have the same one that I have purchased personally, and it works just fine.
That doesn't mean there hasn't been a hardware revision to the design of the adapter, or to the headphones over time as well though.
What it really boils down to is that the apple headphones simply weren't designed for use with anything other than an iphone/ipad. I certainly understand what you are trying to do, but there is likely a better way around this. The general consensus is that the apple earbuds are not the most comfortable or best sound quality anyway, so while you could make it work, the question is whether it is worth it or not. You can get an inexpensive headset that is purpose made for what you want to do, that would have better sound quality, both for the headphones and for the mic. I can recommend some options for that, if you'd like.
Otherwise, what you'd want to do, since there does not seem to be a specific product on the market (at least not any longer) that is compatible, is to make your own.
To do that you'd buy a female minijack connector, or extension cable, what you already purchased would be fine for that even,
keep the headphone side as-is, cut the connector to the mic side, reverse the wires and solder back together, and test. I can't guarantee this will work, because apple products tend to be proprietary and they may have a shared ground or other conductor that is integrated into the headphone side, but this would at least be a starting point.