If you dont know you're in labour then you're not!!! You'll know. The great thing is after an induuced labour you expect the worst in terms of pain, I stayed home til 9cms thinking I was about 4cms lol, with my birth after an augmentation on synt.
Not necessarily! I certainly didn't realise I was in labour with dd#4 until I had my OB appt that afternoon and was at 5cm when he checked me. I'd been very uncomfortable all day but having an irritable uterus I'd been very uncomfortable for months.

I went in to the hospital when the contractions were five minutes apart, even though they weren't painful, because he'd told me I needed to do that, because I'd had quick, easy labours previously. If I hadn't seen him that day, I wouldn't have really realised I was in labour until about two hours before the birth - still time enough to get to the hospital, but not the sort of time I'd like to be moving around on the roads. When I got to the hospital, the midwives didn't believe me that I was in labour - I had to insist that the doctor had said to come in at the point I was cmoing in and that things would happen quite quickly. That's why I think it's important to discuss these things carefully with whoever will be attending the birth.

I also had very easy labours with inductions, so didn't have the same pain expectation that you are describing. Perhaps I'm just lucky