I think it's more the factor of Cheongrok not wanting Jaegyu to feel guilty that he (Cheongrok) cant openly talk about their relationship to even his friends or family to the point of having to get a fake ring just to avoid the subject altogether. He even says that he really wants to brag about their relationship, but he obviously cant in order to protect Jaegyu's privacy. I'm sure Jaegyu would've felt like he was a burden to Cheongrok if he knew the lengths he was going to protect him. As for why Cheongrok hid the ring, it was probably for the reason already listed: not wanting Jaegyu to feel guilty about Cheongrok resorting to lying to people close to him while also suppressing his own feelings.
Is it just me or do I not see a big deal ab getting a fake ring to dispel doubts of a relationship?? Only bad thing was hiding the fact he used a fake a ring, idk why he even hid it