Please recommend a lip and eye remover.

How do you remove waterproof lipstick? I usually used to remove it with oil, but lately it doesn't seem to come off well with that. I'm not sure if cleansing oil just doesn't work well for waterproof products, or if I'm doing something wrong.

I've been hesitating and only using it for pore care, but I wonder if I'm using it incorrectly. Maybe it's because I tend to try to wipe it off in one go, so I even tried leaving it on my lips for 5 minutes, but it didn't work well. Of course, I used it before washing my face with water.

I think I need a lip and eye remover. Can anyone recommend a lip and eye remover that is affordable and gentle?