Are chestnut mushrooms and baby portabella mushrooms the same thing?
Agaricus bisporus is commonly found in two varieties a brown and a white variety both of them have a variety of aliases as you have already found out ... and the name portobella is used for mature fruiting bodies of both varieties. So you can find the brown variety in all sizes from tiny "button" mushrooms over closed and open small and big non-mature fruiting bodies to the big open mature portobellos.

The brown-capped variety will be (at least somewhat) brown through all stages and the white-capped one will have a white cap through all stages. While maturing the gills will turn from white to brown in all varieties, as brown is the color of mature spores in this mushroom species.

Anyway, your assumption is correct - that the chestnut mushroom is in fact a baby (=immature fruiting body) brown-capped portobello mushroom.