She's right at that crucial phase where you could continue gentle measures and work through this patiently for longer, or you could get her on track right away with firmer consequences to clear warnings. It really comes down to your style and how much time you can devote to discipline. If you have one child and a helpful spouse, you have some lee-way. Your daughter sounds very mild-mannered for three. If you want faster results, tighten up. Lots of methods are right as long as you are most concerned about love and teaching.
I never bothered with time outs because it seemed to me they were expecting kids to "pretend" they were getting disciplined. I did exactly as Laurie A spelled out, only I never ever left a venue-but that does work for some.
My toddlers, 3s, 4s had to behave at the park or the store or the concert or the friend's house. We would NOT leave for their bad behavior. They knew the expectations. I made sure they weren't ill or exhausted or hungry. They were prepped upon entry to the locations. They received one quiet calm warning from me to stop whatever bad behavior may start once we were there. BUT, if they then chose to continue the behavior (as all kids do) after being prepared in advance AND given a clear warning, they were removed immediately (bathroom, car) for a pop on the butt and then RETURNED to the scene of the crime to behave with the calm explanation that there would be discipline AGAIN if necessary. I never had to discipline twice in one trip and I never had to do this very often at all for my three kids including the extremely difficult one.
Most of the time at a crowded party or something, once I removed them to the restroom, and they were well in check, worried, knew what they had been doing wrong and expected the worst, I would deliver ONE MORE serious quiet warning at eye level. That solved the issue 99.9% of the time, because they were given a chance to see that I actually could (and would) take action because god knows kids know when they can get away with stuff….It's important to swoop as behavior is BEGINNING and not let them get carried away and then attempt damage control.
I'm a single parent, so I couldn't personally leave the grocery store, Dr office, music lessons.... or reschedule stuff whenever the kids were bad. The biggest perk to this choice is that even AFTER some misbehavior, the activity STILL gets to end well. Immediately after discipline, you can switch gears to finishing the activity, getting back to the birthday party, park, whatever. You're not angry, so you simply go back to the nice event, no grudge. And carry on with your day praising the good and demonstrating to the child how easy it is to stop wrong behavior and get back to having fun. Your brief warning before the next outing will be much more effective if they BELIEVE you.
If you get mad because you've let it get too crazy (kids tend to escalate in phases until they need resetting) and the kids are bonkers and past all hope of listening, it's best to just leave so you can calm down (never discipline while angry or you shift focus from THEIR behavior to YOUR spaz attack), but don't even attribute it to the kids in that case. I've done that a couple of times and told them "I just felt like leaving because you were acting badly. Period." so they knew I was still in charge. Angry lectures and venting will not deter future episodes, it will just get them used to upsetting you with no consequences. If I was NOT angry because I was addressing behavior right away, I would make them behave, not leave.
A note on the swat "only for dangerous circumstances". I'm all for swats to prevent bad behavior when time-outs and logical consequences are ineffective. But if they are inconsistent, as in very rare and only for danger, they may render other consequences less effective-since it sends the message that only in extreme circumstances will that happen. This way the child knows that for things like talking back, defiance, tantrums and other misbehaviors in stores etc, nothing major will happen, so it's very hard to nip those behaviors. If you step up with maximum firmness after warnings consistently in formative years (ages 1 thru 2. 3 is getting up there), you'll rarely have to use more than a warning and then the child's habits can set more easily with more practice being good, and less practice being bad. Once they're older, they have logic and sympathy and self-control, so the consequences change.
I take my three kids to the grocery store EVERY WEEK and have since the birth of my first who is almost 8. That's hundreds of trips to the store with babies, toddlers and beyond. I've only had to swat butts in the restroom about twice. Once in the aisle when I was ENORMOUSLY pregnant (and technically on bed rest but had no one to shop for me) and my 2 1/2 year old looked me in the eye, looked at my belly, calculated my speed, and then ran away after I warned him to stop running. He was both embarrassed and stung (literally, on the butt) when I caught up with him (waddling), and he hasn't run away since, and now he's almost 6. Did anyone around me that day think I was a monstrous abuser? Probably but who cares? I wasn't going to hobble all the way across the store to the restroom with him and have a miscarriage on the account of strangers' opinions. And I wasn't going to chase him every week either. It's the fact that I will calmly, confidently and consistently discipline that enables me to almost never have to. The kids know what they are supposed to do and they know the consequence for deciding not to do it after being warned. The consequences aren't always changing so it's very easy for toddlers to gage. It was instant because I caught the very first infractions. At three it MAY take more repetition...but maybe not since three is pretty sophisticated for some kids.
Let's examine your logical and thoughtful consequences: She misbehaves at park so you leave. Right away or after several chances? Is leaving very hideous? Are your car seats made of wood blocks and nails sticking out? is she taken to hard labor in the gallows? No. She may cry in the car (and get away with a tantrum in that case) or just ride home gazing out the windows at the beautiful scenery on her way to her fun toys in her nice house. If she's getting lectured, she's three so she doesn't care. She tries to run in parking lot? Nothing wrong with that at three-UNLESS she continues after your sharp warning. Once she defies your direction to stop running her punishment is...rolling around the store in a cart? My kids would RATHER ride in the cart! Even if she cries (tantrum) and seems like she'd rather walk in the moment, the ride in a cart is not a punishment. Because it's basically fun and relaxing unless she's melting down which I wouldn't personally allow so I'd have to enact discipline then-so may as well get it out of the way on first offense of running in parking lot when you said no. Because god knows, once they get away with a couple of little things, it only escalates. Acting up at dinner so no dessert or TV? Too delayed for three, and dessert and TV are luxuries anyway. I would make her behave AT the table IN THE MOMENT with immediate consequences and the requirement to stay and behave.
All of the outcomes you list are DIFFERENT so overall there isn't consistency to wrong action, which will make it take longer to sink for her when you are SERIOUS. That is normal. That's the catch with logical consequences. My friends who use them have misbehavior for much longer than parents who stick to ONE effective consequence for all misbehavior. And it's OK to take the longer patient route. Her behavior is very good in ratio to her boundaries. If you say in your ETA you're able to prevent most of this with redirection etc so it's only at home....you've done well with her and this will be nipped in no time if you are effective-or eventually if you go easier and stay consistent. Especially at 3. If she was difficult, you'd be in way hotter water by now. You have a nice child who just needs a bit more firmness if you feel she does. She sounds like my first who has been disciplined like 3 times in her 8-year-old life.
The ways you are disciplining are very thoughtful and you're doing great work being loving and caring and thorough, but sometimes kids need firmer boundaries when logical consequences and time-outs are not stopping behaviors. I wish it wasn't so, no one likes to get tough, but the more effective and consistent you are the less you'll have the issues. The book Back to Basics Discipline by Janet Campbell Matson is really good for this age.