Sounds like the first basil you mentioned MIGHT be semi-new Pesto Perpetuo, which is a non-flowering basil which can be overwintered indoors, given the right conditions, I'm told. I grew it for the first time this year, and I love it. The taste is wonderful and the plants get really huge. Besides, it's a beautiful plant with it's variegated leaves. I have four very large PP plants, and I am going to TRY and bring them inside this winter. We'll see.

You MAY have some success with Sweet Basil for a while indoors, but it's just not meant for inside. I've never had much luck keeping sweet basil alive indoors. Even those small kitchen pots of basil you see everywhere don't do well after a while...they just don't get the sun nor have the room to do their thing. Besides, it naturally dies during the winter and does not come back in the spring, as do some herbs.

I just replant every year. It's so fast-growing, there's no need to keep an old sweet basil plant around. And during the winter, if I run out of what I've "put up" in the feezer (put in the food processor with a little water and then frozen in cubes in ice trays), I have been known to buy the large containers of sweet basil at the supermarket, in a pinch.

But with Pesto Perpetuo, you can bring it inside. But it will need plenty of light and you'll need to keep it pest free. It will need some food throughout the year and of course, drainage is hugely important.

JMHO.