I dont even consider HKIII as a serious matter. It's just his first spell for me.
Possibly you could say that Catterick had it over HK in that he built two titile winning teams whereas HK had basically the same core he used for 85 and 87; however, HK gave us European stature as winners of a trophy.
HK for me.
This is an argument for another thread but Catterick built the greatest Everton side of all time imo.I think from about the early 60's to the 1970 league title that Everton team collected more league points over a 10 year period than highly rated Man United Red Shyte Leeds and Man City teams.