I'm fairly certain, as well, that this is a Fukien Tea. Propagating through cuttings is possible, so is air-layering. Fukien Teas, historically, do extraordinarily poorly when kept indoors. However, they can not be kept outside during New York winters either. Unless you live in a zone conducive to keeping the species, it will have to spend a good amount of time indoors. For Fukien, this means six months out of the year, or more. I would not be comfortable keeping one outside until after Mother's Day, and I'd bring it in before October. The keeping inside is what makes them difficult. They need higher humidity and prefer to NOT dry out at all, ever, IME. I've had a lot more luck with this (and similar species like Serissa and Sageretia) tree by ignoring the traditional advice to use low-organic, fast-draining soils. Simply put, there are many ways to do things, usually, and to ignore non-traditional methods in favor of traditional ones that are proven to fail thousands of times over is... well... silly.
So, if your tree starts dropping leaves, either set it up in a terrarium, or repot it into a mix that is at least 50% potting soil. Seriously. My first successful overwinter, ever of Sageretia, Ehretia, and Serissa all came from ignoring traditional advice.