Yes, there is a difference between Chinese "battery grade"...

Yes, there is a difference between Chinese "battery grade" bottom end 99.5% purity which attracts the low end spot price, and 99.9-99.99% battery grade hydroxide that will go in a Tesla.

Here's a great article on the difference by Prairie's Ian Ireland.
https://www.labmanager.com/big-picture/lithium-ion-battery-production/the-mining-and-refining-challenges-to-produce-high-purity-lithium-26230

Q: What are the key needs and requirements of these Li customers?

A: The key needs for LIBs are high purity Li salts—either lithium carbonate or lithium hydroxide monohydrate (LiX). While the current standard is 99.5 percent pure Li salt, battery manufacturers really want at least 99.9 percent pure, and are interested in getting 99.99 percent, or even 99.999 percent pure product.Low impurity rates in the Li salts are critical to battery performance and safety. Impurities, like sodium, have led to battery failure, overheating, and fires.

It seems that the hydroxide path for Prairie had been in the works for years according to this 2021 video.

Yes LF, battery manufacturers will pay a higher price for 99.9% purity than they will for Chinese 99.5% purity.