Take things slow and be careful. You don't want to do something that you might later regret.
There seems to be a couple of different aspects to your problem that you can deal with separately, rather than trying to solve the whole thing in one swing.
The job title
It seems from your question that there is a company-recognised procedure with job titles. This is a great place to start. Inform your co-worker (in email) that you want her to use your customer-facing title in all correspondences going forward. Start off by making a simple, polite request.
If that fails, follow up with a second, more firm, attempt that you have to insist that she uses your correct title. If this second attempt fails, take it to your manager - no need to follow up further, you have been reasonable enough already at this point. Depending on what your manager says, you may need to escalate it even further if you think it is worth the effort.
Do this first, and await the outcomes before you take any further action regarding her trying to be your manager.
The claim she acts like you manager
I will start by saying that your examples are not good examples. They do no support your claim that she is trying to be your manager, and although others on this site seem to think they are, they definitely won't hold up as any substantial evidence to support a formal complaint against your co-worker.
Keep in mind that "co-worker gossip" isn't your friend. Other people saying "she sounds like your boss" does not prove that she is, nor that she is trying to be.
If you have better examples that you don't want to share, but you feel do support your claim, then it is on you to decide if you feel they will justify and support a formal complaint.
Having said that, the import thing to consider is what you feel you have to gain from making a complaint, and are your superiors likely to agree with your concerns.
Let's assume it's true and she is telling people she is your manager. Does that affect your ability to do your job? Does it affect the companies image and revenue? Does it have a negative impact on the vendors/customers? These are the concerns that the company will have if you raise a complaint, and these will be what motivates them on how they respond.
And perhaps have a bit of self-reflection and ask yourself why do you even care? You know the truth, your superiors know the truth, and by the sounds of it your other co-workers also know she isn't your manager. Don't let it get to you.
If however, there are knock-on effects to your ability to work or to vendor/customer relations, then perhaps you may need to take further action later. But for now, start with addressing the job title concern and go from there.
Why I think your examples are bad
In case you are wondering why I think your examples are weak here is why. I feel this may be important to help you see a perspective as to why you might be reading too much into them.
'I am cc'ing in our Specialist II who can shed further light on this'
This is essentially just saying: "I am not the best person to help you with this, I will pass this on to our "Specialist II" who is better equipped to help."
If anything, that is suggesting she isn't capable of helping and you are actually more capable. I would take that as a positive to be honest.
'You may contact me and the rest of the Duty Manager team if you need further guidance'
This is simply saying: "If you have any further concerns, please let us know."
Remember the customer contacted her first, so she has to be the one to "take control" of this communication. She can't very well turn around any say "I'm not the right person for this, don't email me again". That would be very unprofessional.
I think somebody commented this already but: put yourself in her position, what would you say in those emails?