This isn't necessarily a huge deal, and you are blowing it out of proportion (given what you've reported here). Relax, understand what mistakes you made, and accept that you made them.
Let's get a few things out of the way, as there is a flurry of information in the post that may or not make you feel better about the situation but does not help you out of this position:
- You absolutely did something which can, apparently, get you in trouble (asking for the phone number)
- You've acknowledged that that you did this, and were aware at the
time that it was not appropriate
- Your description of events is thin, and does not mitigate the mistake
- You seem to be in a highly defensive posture, but haven't suggested
any potential consequences that you've suffered (or been threatened
All other details are secondary. If your employer wants to reprimand you, they have a sound basis for doing so. There is not a single thing written in the question that makes the mistake better or milder for you, and trying to minimize events will give the impression that your judgement is still not good enough to navigate similar situations in the future.
Some things that don't matter very much:
It is irrelevant if you've ever done something like this before or not. The question doesn't suggest that anyone is accusing you of serial bad behavior. If employees are supposed to ask customers for their phone numbers 0% of the time at most, the argument that you haven't asked for phone numbers 90% of the time is worthless. It's the time you did ask that matters.
It is curious that you assert that there is no rule against asking for phone numbers, but also assert that you did something wrong by doing so in this case. If it's not prohibited, it's not clear why you would be concerned. "It's not against any rules or policies" is a solid, affirmative defense (if it's true), and it wouldn't matter if you ask every customer for their phone number. Especially if you're aware of other instances of people from your company doing it. That you are concerned anyhow seems odd, though I don't doubt there are potentially valid reasons that you might be (I just don't know about them here).
Your description of events doesn't help you very much. It's not that it's incriminating or anything, it simply does nothing to rebut what you've heard about the complaint. The worst parts are that you went outside for a reason you don't recall, conveniently happened to run into this woman, talked to her but don't remember what you said, and asked for her phone number. This is all consistent with the complaint.
Your guesses about what the secret shopper was doing and why are also not important. Presumably she understood her job and was doing it in accordance with her employer's requirements. It is unlikely, though possible, that she decided to risk her job in order to mess with you by knowingly filing a totally false complaint. It easier to believe that there is some truth to what she's describing (even if it's not 100% accurate) than to believe that she made all of it up for no apparent reason.
Your assumption that she seemed more interested in you than in your products is also unimportant. Even if part of her job was to "simulate" a customer displaying some interest in you, a claim for which we have zero evidence of any kind, then she discovered that you at least might respond in as you did. It wouldn't really be different than if she were testing whether or not you would give away free merchandise to an attractive woman who asked for it-- even if she prompted you, you are still supposed to follow the rules. Advancing this claim only suggests that you think, under some circumstances, asking for a customer's phone number is OK. That doesn't mesh with an acknowledgement that it wasn't.
Something that does matter:
If there is internal security footage in the store, try to get a copy of it. The secret shopper's claim that you were following her around may be more serious than asking for her number, and camera footage is the only convincing evidence you're likely to come across. Bear in mind that, even if you can demonstrate that that particular claim is not true you are not "innocent". You have acknowledged doing something you should not have done.
Your specific questions, in order:
She kept repeating "be careful". What does that mean?
You were not very cautious when you asked for this person's phone number, and it's led to a problem. You should make efforts to be careful in the future, to avoid future problems.
Does quitting make me look guilty?
Maybe, but I don't see that it matters. You aren't being targeted for a civil suit or anything. It seems like the consequence you're worrying about is some kind of formal reprimand or losing your job, but the complaint itself probably doesn't exist outside of a note in your HR office's file.
Is there a way I can still work part time at this company and would it be a good idea?
I don't see why not. They haven't fired you so far, and indeed there seem to have been zero actual consequences at all. You'll know your employer better than us, but it doesn't seem like your job there is ruined. As for whether or not it would be a good idea, again I don't see why it wouldn't be (as things stand now). Your call, either way.
Is there anything I should or can do at this point?
Get the video footage from the store (if available) to demonstrate the one (potentially) verifiable claim still at issue.
Shift your rhetorical posture. Your current position seems to be that you feel you deserve to suffer no consequences of any kind, despite admitting some fault. That will be difficult to maintain, especially if you want to keep working for this employer and they impose any sort of punishment over this incident. Own up to your mistake, accept that there may be consequences for it, and try to move on. Don't endlessly relitigate this issue trying to find reasons your mistake wasn't so bad, or wasn't your fault, or any other kind of minimization.