SMJ - it's not as complicated as it may seem.
Panzer is a shortened form of Panzerkampfwagen (usually abbreviated PzKpfw), which translates to Armored Fighting Vehicle. All the German gun tanks (as opposed to command or observation tanks) were called PzKpfw I-VI. The Panther was Panzer V and the Tiger I & II were called Panzer VI.
All German military vehicles were given a Sonderkraftfahrzeug (SdKfz or special purpose vehicle) number which was their official designation. That might be the best way to keep track...
SdKfz 171 Panzerkampwagen V 'Panther'
SdKfz 173 Jagdpanzer V 'Jagdpanther'
SdKfz 181 Panzerkampfwagen VI Ausf. E 'Tiger'
SdKfz 182 Panzerkampfwagen VI Ausf. B 'Tiger II'
SdKfz 185 Jagdpanzer VI 'Jagdtiger' (88mm PaK 43 L/71)
SdKfz 186 Jagdpanzer VI 'Jagdtiger' (128mm PaK 44 L/55)
Sdkfz 267/268 Panzerbefehlswagen V 'Panther' (Panther command tank)
Sdkfz 267/268 Panzerbefehlswagen VI 'Tiger' (Tiger command tank)
Vehicles other than gun tanks based on Panther or Tiger hulls have a different purpose and a different number, or no number at all in the case of the field-conversions, the Tiger I-based Bergetiger and the Sturmtiger.
SdKfz 179 Bergepanther
Bergetiger
Sturmpanzer VI 'Sturmtiger'