Introduction going poorly

I adopted a cat while I was away at uni. I’ve had her 11 months and she’s a little over a year old. My family has a cat and she is a little over 4 years old (we have had her almost 4 years). My cat is very friendly, likes other cats and people, and really affectionate. Our family cat is really shy with any non-family members and not as affectionate, and doesn’t have much experience with other cats.

I was home in December and brought my cat with me. I kept them completely separate (litter, food, and water in my room with my cat) but I have a big family and a mishap happened with people coming in and out of rooms. Our family cat was REALLY territorial and did not like the introduction of a new cat. She fought with her when she managed to sneak into my room and my cat was really afraid of her to the point of always hiding under my bed when she NEVER hides. Because of that I moved to a different room with my cat and made sure our other cat wouldn’t be able to sneak in. I took her back to uni with me but it’s been 7+ months and I moved home permanently after graduating.

I again kept my cat in a completely separate room making sure the family cat couldn’t see her but she can smell her and she does the same thing. I thought the distance would help but they remember each other. She waits outside my door so she can try to get in and fight my cat. Neither of them respond well to scent swapping. My family cat has PLENTY of space to wander and her own food and water bowl and litter in a separate room. Am I doing something wrong? What should I change so I can move to the next step?