I am not an expert on timbers, but my understanding is the following. The term rosewood is used as a collective term and has more to do with marketing than with botanical classification. The English term rosewood refers to palisander, which is the trade name for various precious woods of the genus Dalbergia. This includes Sheesham (Dalbergia sissoo), which is native to the Indian subcontinent and cultivated in large plantations. The wood referred to as "Rosewood" by Indian manufacturers usually relates to Indian Rosewood (Dalbergia latifolia). Since 2016, all species of the Dalbergia genus have been listed in Appendix II of the Washington Convention (The only exception is Brazilian rosewood (also called Rio rosewood, botanically Dalbergia nigra), which is a highly endangered species listed in Appendix I of the Convention and subject to even stricter controls), which means that commercial trade is only possible after a clearance procedure by the exporting country. As far as I know, the Indian government (in this case representing the exporting country) is not limiting the use and export of sheesham and rosewood. Regarding "bud rosewood", it is slightly different. The wood so designated does to my understanding not even come from a Dalbergia genus, but is African Padauk (Pterocarpus soyauxii), which is currently not subject to any trade restrictions.
Here are examples of (1) Sheesham, (2) Rosewood and (3) Bud Rosewood:
I've been looking at buying a set which comes in a couple of variations. The white pieces are boxwood, and the dark pieces are available in ebony, rosewood, bud rosewood, or sheesham. I was under the impression that these woods are banned from export/import, except for ebony. While I know there is a lot of confusion with naming practices, and that some of this is happening intentionally for various reasons, I can't figure out what is going on. If I buy one of the rosewood or sheesham sets, am I violating endangered species laws? Or are the woods being used permitted species that are actually not what we historically have thought of as these woods? Finally, if it's ok to purchase these, why would the "rosewood" command a significantly higher premium over "sheesham" as they seem to me to have pretty similar appearance and physical characteristics?