I was induced with drip with my first baby and had a planned homebirth with my second, so I didn't need to worry about getting to hospital for my second. The pain was very different in the latter stages of labour, contractions were much stronger and closer together whereas normal labour was gentler. My waters didn't break until second stage with my second unlike my first that ruptured 2 weeks before my induction, so every labour is different and unfolds in a unique way. Technically your labours were actually augmented not induced as you had already started labour (dialating is actually labour), your care providers decided for whatever reason that your labours needed to be 'sped up' with a drip or rupturing your membranes. Augmented and induced labours are often short and because your labours were both augmented then it is difficult to determine how long you will labour natually this time around. Sounds to me like you are making a really good choice to birth at a birth centre and my suggestion to you would be to educate yourself about the signs of pre-labour/early labour and active labour so that you are very self aware of where things are at. I would suggest that you stay at home until you feel you need to go into the birth centre, if you are self aware then you will intuitively get this timing right. If you are concerned about missing the timing then make plans for an 'unplanned homebirth' so that you don't panic. You are better to birth at home with an ambulance on the way than in the back seat of your car or the birth centres steps.
I am sure everthing will be fine.