Both cpus are showing as LC versions. Weird. Have tried installing the libraries you mentioned (from Phase 5 B1260 archive) but is not doing any good.
At least WhichAmiga reads the 68060 processor configuration register directly (PCR). Full 68060 and 68LC060/68EC060 have different bit patterns there.
bits 31-16 (upper 16 bits):
%0000010000110000 full 68060
%0000010000110001 68lc060 or 68ec060
Thus, neither of the cpus is full 68060.
Also, it should be pointed out that WhichAmiga works regardless of the 680x0.library configuration (that is: it tests for 68060 directly, bypassing the libraries).