Because you didn’t change anything other than the phone, Mark is correct, all you needed to do was log into the Blink app, with your usual credentials, but clearly, from the lights you have described, on your sync module, you have already pressed the reset button. If you hadn’t, the whole system would have continued to operate, as indeed it does when your phone is off. It has to work, in the absence of a phone, in case your battery goes flat, or whatever else might be wrong with your phone. Wouldn’t be much of a monitoring system, if it stopped recording when your phone was off.
That having been said, if you have reset the sync module, it’s kind of moot now, other than, you might be able to rescue it with your old phone.
Because you are more familiar with the workings of your old phone, why not rejoin the sync module to your system on the old phone, assuming you have a SIM card that will work with it. You did it once, so you should be able to do it again. Once done, and the whole system is working, you can then simply open the Blink app, on the new phone, and log-in with your usual credentials, and everything will be there as it was on your old phone.