If it was a little later than 12th (sounds totally possible) then your dates make complete sense. Bear in mind in the first tri scans can be a week out, it's common to measure "behind". I'm measuring 4 days behind. How long are your cycles?
Assuming a 30 day cycle you would have ovulated around 30th September..You'd only be 5dpo the first time you tested, which means the negative result makes sense...and you'd be a healthy 12dpo a week later...When you're likely to get a positive.
That means you'd have been less than 4 weeks pregnant on 10th October.
Let's assume you were 3+4. That puts your scan that said "around 5 weeks" bang on time...The difference between 5+0 and 5+4 is really variable...Some only see a sac and others see a heartbeat...Basically, they said "around 5 weeks" because they can't tell.
Which means you're now around 7 weeks pregnant, making your 6 weeks 3 days scan fine too. Remember, a week's error margin either side. 6+3 measuring scan can easily be a 7 week pregnancy.
It's very tricky to get your head around but I hope this makes sense!
Basically, because you can't be sure when you conceived, I would relax, take the scan dates as your dates as and when they come (they will always change!) and enjoy early pregnancy. A heartbeat is a massively promising sign. In fact once you've seen the heartbeat any risk drops vastly.
I think you're anywhere between 6+3 and 7+3 right now and whichever it is, all is healthy!