I think it's nothing to do with you. Just her. For me, I can't let my kid cry. It breaks my heart to hear any baby cry and not be able to comfort them. My friend has a kid my daughters age and CIO worked wonders for them! It's just a personal thing! Your mom is probably like me where it like hurts to stop yourself from comforting them. You need to remind her that this is your child and not hers. And even though she may not understand or agree, she needs to respect you as a mom and trust that you know what's best for you and your baby.

I hate cry it out method so I put my baby crib next to mine and hold her finger through the night

It's hard to hear a baby cry for some people, it's like unbearable for my MIL to hear a baby cry. Just keep reminding her that it's what's best for your son!