She has been tested and has ADHD and is on medication for it.
Ritalin is NOT an ADHD panacea. It manages some of the symptoms, but doesn't teach kids to make good decisions or manage their impulses. You can't rely on meds to change their behaviors - but the meds can help the people involved.
We tell A__ that his meds are to help him learn to manage his ADHD without meds. But the work of teaching him to manage it is up to us.

The one thing that we are really having a problem with is her mouth. She is completely rude!
Most 9yos have a bit of lip. It's pretty normal for a kid that age to have some rude moments.
When A__ gets mouthy, we say something like "Try that again without the attitude" or "Please say that again in a more polite way"

We both talk to her about it but her response is "people need to change for me, I do not need to change for them".
It sounds like one of two things is happening:
1) Someone (maybe BM? maybe her doctor? maybe her teacher?) has given her the idea that she is "broken" and that is an excuse to not try to act like a proper person.
or 2) The talking isn't getting through the "ADHD noise" and into her brain
I don't know how you would deal with #1.
But for #2 you need to revamp how you talk to her. Lectures do not work on ADHD kids. With A__, if we talk at length to him, he glazes over and then interrupts with inane 9yo blather ("Andguesswhat on the computer blah blah blah"). If we keep our talking to <30 seconds, and then ask him a question or ask him to explain what we just said, it actually sinks in and he retains the point of the conversation.

Her and I really but heads a lot and I do not know how to fix this.
I love A__ to itty bitty bits, he loves me too, and we have a great relationship. BUT this is work for us both. We have an agreement that we both have to put the effort in and then we both get to enjoy a good relationship. My "jobs" include providing care for A__, and my "perks" include spending lots of fun time with him. His "jobs" include listening to me, and his "perks" include spending lots of fun time with me.
We have to revisit this deal every few months, and it took some building to get there, but I would recommend you try to make an agreement like that with her.

As she has gotten older she has gotten worse with the attitude and mouth.
That's pretty normal.

Her mother recently got married. I have asked my SD if she treats her Step dad as bad as she treats me and she will tell you no.
We're on the other end of that - we're about to get married, and A__ and I get along well. But BM has a BF who A__ has problems with.
A__ and I get long because we CHOOSE to, and FDH supports us having our own relationship. BF and A__ have not made that choice, and BM doesn't allow them the freedom to develop their own dynamic.
I don't know what the differences between BM's and your house, but what can you think of that is different? Could any of those things be causing the disparity between how she acts between you and SF?

She thinks the world should revolve around her.
Of COURSE she does! She's 9! 9yos are egocentric.

She complains that she is the only one that gets put in time out and the only one that gets yelled at. I am harder on my own children then I am on her.
We hear that from A__ too and we don't have other kids!
One thing we found with A__ and time-outs: he needs to be the one making the choice to behave or the time-out has no effect. So if he goes to his room "for ten minutes" he comes out in the same bad mood as when he went in. If he goes "until he's ready to apologize/be polite/be nice/etc" then there is a change in him when he comes out.
Once he sat in his room for an hour and a half. Sometimes he comes out two minutes later with a smile and a new attitude. Sometimes he sobs his eyes out or yells for ten minutes and then he calms down and eventually appears, much improved.
I'd really recommend trying that on SD when she goes for time outs. It makes a big difference.

I don’t know how to deal with her attitude and mouth.
I don't know of anyone who DOES know what to do about their kids all the time. We're fumbling through things with A__, without all the answers, trying things that don't work until we find what does.

Plus to be very honest, I do not trust her. She will convince my children into doing things they are not supposed to and then they get into trouble.
Most kids try to get their siblings in trouble.
But please try to learn to trust her. This will be work for you both, but A__ has learned that, with us, if he is honest or "tells on himself" we might be annoyed or frustrated, but he won't be in trouble. So it's OK not to eat your lunch at school (the meds kill his appetite) but it's not OK to throw your lunch out and say you ate it.

Last summer she tried to talk my daughter into eating rabbit pellets telling her they were berries. I have told my husband that if something doesn’t change soon, we will be bailing her out of jail when she is 15.
A friend's older sister made her eat a goldfish and is now a successful 32yo businessperson. She hasn't spent any time in prison at all.

The big problem is, she does not live with us and if you ask anyone about her they say that her problems are cuz of her ADHD. I do not believe this; I think it is lack of discipline at home. How can you fix the problem when you only have her part time?
ADHD is just an obstacle that makes problems bigger and more difficult to overcome. It is not a reason to not try. Nor is it the cause of all the problems.

Short answer: Slowly and with a LOT of work.

Long answer that has worked for us: We started off by making a set of "rules for our family" that has rules about being nice and telling the truth. We went through the rules with examples of following and not following, and then started invoking them as needed. That way, he gradually learned the standard of behavior we expected and we constructively helped him find how to meet that standard. The first few hours of A__ coming over were never good, but after he adjusted to having to live up to expectations, he'd get right into the good habits.

We coupled the new rules and expectations with ADHD things like a timer for the computer or until bedtime and so on. We had to use the timer for practically everything for about 4 months, until he got a bit of a concept of time and learned a few time-management techniques. Learning time-management was harder for A__ than for other kids, because of his ADHD, but he did it. We also used lots of other little ADHD management things (checklists, routines, etc)

Learning good behaviors is harder for kids with ADHD because they can't give a "neurological self-reward" just for making a good choice. They need to get an external stimulus that gives their brain that reward until they internalize the behavior into a habit or into their nature. So we had to give positive feedback for EVERYTHING (I got so sick of high-fives) for MONTHS. "You hung up the tea-towel when you were done drying dishes! Yay!! Gimme a high-five!"
Nowadays we still give a LOT of positive feedback, but it's different: "Thanks for putting the tea towel away" instead of the whooping and cheering.

Even once A__ learned to meet behavioral expectations, he still did a lot of things to get negative attention. We talked with him about "good or bad attention" and what each is. So A__'s definition is "good attention" is doing something that someone might give you a compliment for... and "bad attention" is doing something that someone might tell you to stop. We brainstorm with him on ways to get good attention. We ask him "Are you looking for good or bad attention?" or "Can you think of a way to get good attention instead?" and it has gradually worked.
I'd really suggest this if she seeks negative attention, as most ADHD kids do. See, their brains take all attention as positive because good or bad, it triggers the same chemicals in their brains. So you have to stop giving feedback to negative and give A LOT to positive, until looking for positive attention becomes a habit or part of their nature.
It's slow going, but trust me that it can be done.

We stopped engaging in drama. If A__ screams "I hate you! You're the worst dad in the world! I wish I was dead!" because he was asked to brush his teeth, we respond with a quiet "It's too bad you feel that way, because I love you and am glad you're alive. Please go brush your teeth now."
If we respond with yelling back, or trying to soothe his anger, he FEEDS on it and the behavior escalates. He does it again the next night when he is told to brush his teeth.
I advise against acknowledging outbursts of drama.

BM has commented to us that A__ is so much better behaved and that she loves how good he is now. But BM does have some structure for him. Her mom, on the other hand, has no rules or expectations and FEEDS bad behavior like you wouldn't believe. So if we pick up A__ from BM's, it's a short changeover to the behavior we expect. If we pick him up from GM's, it's a long adjustment.

Anyhow, it CAN be done with a kid who isn't with you much, and is usually in a less constructive environment. It's just longer and harder.
But believe me, it's worth it.