Actually, Steve's Dad's name is BOTH John AND Jack & either/both would be correct in referring to him in canon. I guess you're referring to that in your post?
On his headstone, which we see at the end of ep 104, Lanakila, when Steve & Mary are having the picnic by their Dad's grave, it says "John McGarrett" along with his dates of birth & death (the date of death is the same as the date of the S1 premiere... September 20, 2010); so John is is given name.
His nickname is "Jack", which is a normal nickname for men named John (President John F. Kennedy was also referred to as "Jack", for example). That's also the name Chin--who'd been trained under & partnered with Steve's Dad at HPD, so he'd have a familiarity with him--uses for Steve's Dad in ep 102, Ohana. Chin says something like "Jack McGarrett considered any day Steve McGarrett played at Quarterback a day off" (maybe not a direct quote from the ep, but at least a paraphrase of the line), when Steve expresses surprise at Chin's previous revelation that he & Steve's Dad used to come watch his HS Football games.
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On his headstone, which we see at the end of ep 104, Lanakila, when Steve & Mary are having the picnic by their Dad's grave, it says "John McGarrett" along with his dates of birth & death (the date of death is the same as the date of the S1 premiere... September 20, 2010); so John is is given name.
His nickname is "Jack", which is a normal nickname for men named John (President John F. Kennedy was also referred to as "Jack", for example). That's also the name Chin--who'd been trained under & partnered with Steve's Dad at HPD, so he'd have a familiarity with him--uses for Steve's Dad in ep 102, Ohana. Chin says something like "Jack McGarrett considered any day Steve McGarrett played at Quarterback a day off" (maybe not a direct quote from the ep, but at least a paraphrase of the line), when Steve expresses surprise at Chin's previous revelation that he & Steve's Dad used to come watch his HS Football games.