For the first few charging cycles the new battery acted like it was not taking a full charge, but then settled down to behaving like I had hoped -- charging up quickly and then running all day with capacity to spare.
But just yesterday the phone started acting like it was NOT charging. The little lightning bolt icon came on when the phone was plugged in, but the percent of charge did not increase -- it slowly went down.
There did appear to be substantial charge in the battery, despite it being 'in the red'. It took a very long time for it to completely discharge.
Going by comments found online, I allowed the phone to completely discharge, then plugged it in overnight. It was up to about 70% charge when I checked it in the morning. I did a system restore on the phone -- an option in iTunes. It wiped the entire system then rebuilt it from the iTunes backup files. Dunno if that helped, hurt, or had no effect. But so far today the battery and charging have been back to behaving normally.