Neem and Chinaberry Name Confusion

Will you please explain to me the following:
In your online catalog I found "neem tree". Your description is: Azadirachta indica (Melia azadirachta)
When I searched another site (Indian site, which sells neem tree products), their FAQ page says:
1. What is the Botanical Name of NEEM TREE?
AZADIRACHTA INDICA A JUSS
2. Is it different from MELIA AZADIRACHTA ?
Certainly. The Fruits of MELIA are in fact toxic.
My question is: what kind of neem trees do you actually sell at Richters?

We sell the correct plants, Azadirachta indica.
The confusion lies in the fact that an old name for neem is confusingly similar to the name of another closely related plant, the chinaberry.
The currently accepted name for neem is Azadirachta indica. Previously it was known as Melia azadirachta. However, chinaberry is Melia azedarach which is different spelling, but similar enough to be confused. Richters carries both neem and chinaberry. They are easy to tell apart because the leaves are quite different. The bark of chinaberry is medicinal, being used as an emetic and to expel worms.
The author on the website you consulted is confusing the old name Melia azadirachta (neem) for Melia azedarach (chinaberry).