Accounts of Lydian campaigns (Pteria & Thymbra) quote Lydian cavalry as armed with long spears. Is there a case for changing the cavalry in their list in their list to Long Spear rather than SSp; (maybe at cost of Melee Expert ??)
Not responding on the classification, but you might want to look at this topic (https://mortem-et-gloriam.co.uk/smf/index.php?topic=3011.msg20695#msg20695), which discussed some points on the differences between CL, LSp and SSp.
Best,
Antoine