There is no distinction between the two in terms of regulations (all the ones I've seen are explicitly calling both lithium ion and lithium metal batteries).
>[T]he CR2032 battery is a Lithium-manganese dioxide battery (LiMn02). It is composed of a Mn02 cathode and a lithium anode. The device is specified for a 225 milliamp hours (mAh) and typically operates over a temperature range of –20 ºC to +70 ºC.
You will find some alkaline versions available from some vendors, but it's not the norm.
Now there may be a chance that Lufthansa didn't mean to include all types of lithium cells or batteries, but the vague wording doesn't seem to suggest that.
Lithium batteries of any kind have the same regulations in aviation. Here[0] is a link from the US FAA, but you'll find similar at ICAO or any other civil aviation authorities.
Both - when installed in a device - can be transported in checked luggage though: https://www.iata.org/contentassets/05e6d8742b0047259bf3a700b...
IATA actually has a document specifically addressing tracking devices: https://www.iata.org/contentassets/05e6d8742b0047259bf3a700b...
My reading of that document indicates that an AirTag should be fine, as low-powered wireless communication is allowed (page 9, item 3).