For Priority Pass at LHR, only the arrivals lounge at T2 doesn't have the "smart casual" language. I suspect that's an unintentional omission, however, as the arrivals lounge at T4 does have that language.

That said, some of the lounges on the list also mention "no football shirts". That makes me think that their definition of "smart casual" is pretty lax. The last time I flew in and out of LHR I wore a collared shirt and regular (not dress) pants just in case and didn't have problems.