From what you have said, you come off as a beginner.
When people in ads say that their horse needs an intermediate rider, they either-
1. Know that their horse can be a bit difficult to handle, he might be a bit difficult to put the bridle on, might not lead 100% quietly, might not load well, could be fast under saddle, might shy, could buck once in a while from excitement, etc.
2. The seller could think that their precious baby has a lot of potential, and thus wants a intermediate/advanced rider to help them "reach their potential".
3. Just doesn't want their horse to go to a beginner home.
You have to think for yourself what level you are. In my book, any one under 18 is a beginner/ advanced beginner. I consider myself an advanced intermediate rider, and I ride Grand Prix tests in shows. But I couldn't train a horse fully to do Grand Prix movements, I can just work on the movement once it has been schooled a couple times. My trainer/boss wants me to master as much as I can, and then working on learning how to PROPERLY teach piaffe, passage and spanish walk.
So, on to you. What can you do besides ride the standard gaits? Anyone who has had 3 months of good riding can do all you have said you can do. Can you control a horse that has bolted? Can you ride a buck/ rear/ hissy fit and then get the horse back under control and continue working? Can you re-seat yourself after you have lost your balance/ stirrups/ reins? Do you feel comfortable lounging a horse? What about a crazy horse bucking and rearing on the lounge? Can you catch a horse that doesn't want to be caught out in pasture? Can you handle a horse that is freaking out on the end of a leadrope? Can you do basic first aid? What about more complex first aid? Do you know when a vet needs to be called? Can you tell when a horse is lame? Do you know how to feed? What about amounts of feed? Supplements? Can you deal with/ fix bad ground manners?
These are the things that truly determine the level of a horsewoman/man. More importantly, can you do all the above mentioned things while keeping a level head and not losing your patience?
If you said yes to almost all these questions, then you are an advanced beginner/ intermediate.
NOTHING EVER should EVER be determined by how long a person has been riding. How long you have been riding isn't worth horse ****. You could have ridden you whole life on half dead school horses, and that isn't real world riding experience.That reminds me, have you ever ridden a horse besides a school pony?
Just go see all horses that interest you. You will be able to tell almost right away if you can handle the horse. If you can't or you feel you are in danger, you can always just leave. I'm sure the seller would also express their concerns if they felt that you wouldn't be able to handle their horse.
So to summarize, I think you would be fine to look at horses that are both intermediate and beginner, however I would personally aim for finding a nice school master, that has done everything, and then you can have fun moving up the levels in your riding, and showing.
Also, don't listen to the barn bitches. They usually are just that. Believe in your abilities, and have fun horse shopping.