I would suspect that the USB-Etherner adapter is not compatible, http://chinagadgetsreviews.blogspot.com/2013/06/rj45-ethernet-network-lan-to-usb .html since the spec's don't list support for OSX.
The specs listed are: The LAN Network Card support Win95 OSR2/98/98se/ME/2000/XP/NT3.5 or above, Linux.
All that info you have about built in modems is false.
Thanks for replying - I should have specified - internet on my macbook air works normally. I just can't share it to my phone.
and also, strangely, my adapter doesn't look like the one shown in the webpage you have given, its label says its the same model but mine looks like the apple brand adapter in white with a cable.