I’m very familiar with this. Cell phones use lipo cells and those are specifically which are mentioned. Those are quite dangerous however ones in cell phones are well under 100Wh. Packs over 100Wh are considered dangerous enough to be a threat to air travel hence the regulations. The battery in the TEMPEST is a completely different technology.
This ultimately is up to the carrier and the shipper. When required the SHIPPER must disclose what’s in the package and whether or not it cannot be flown. I’ve received two replacements already and returned them and they did not have any labels on the box indicating lithium batteries were present. When I ship or receive cell phones, tablets, et al we DO in fact have to put these labels on the box. Same goes for loose cells (ex. 18650).