To each their own, but Hellbound was the much much more intriguing film of the 2 even if the execution wasn't very slick ( at least in the last 1/3rd ). Hell on Earth is one of those movies that ends and you're just kinda like "welp, I guess that's it".

I don't find this film to be anywhere near as bad as most people say it is but an improvement over the second film? To each their own I guess.
Horror_Metal

Agreed. I was extremely disappointed by "Hellbound" (overly gory and yet also extremely cheesy). This one was a huge improvement.

From what I understand, they did not understand the popularity of Pinhead at the time. It was instead Julia who was supposed to be the Big Bad of the series, but the actress wanted out as did the actor of the doctor after Hellbound.
As you said, H3 is really an attempt to fix things from the second.

To be fair, he wasn't Pinhead at the time of his death.
HI-F___ING-YA
Nicholas Cage Deadfall
Films 2015:
www.imdb.com/list/ls073224289/

I honestly thought the first two films were gorier than this one. I wouldn't rate it higher on account of the violence. I'd rate it higher because the second film was very flawed and most of what it got right came from directly copying the original.
Frank's Hell seemed more like a Family Guy joke than a legitimate form of torture. Channard wasn't very interesting. Making the Cenobites human took some of the magic away from them. Mostly though, the fatal flaw of the second film was in letting Pinhead lose to Channard. It'd be like having Freddie Krueger killed by an evil doll and then returning to business as usual in the next film.
To be fair though, they didn't intend to keep making sequels. As far as they were concerned at the time, Pinhead was dead. I still think that's a poor excuse to have him killed off by a less impressive antagonist.

You mind if I ask why you disliked the examples you listed(Frank's Hell, Pinhead being human, Channard killing the cenobites, Leviathan being a "goofy giant box")?
I think making the cenobites former humans and the introduction of Leviathan added to the mythology and the only thing Hellraiser III contributed was fleshing out Pinhead's backstory. Otherwise, it was an obnoxious, cheesy mess of a movie that was made to appeal to gorehounds who wanted Pinhead to act like a combination of Jason and Freddy. It was as if it only exists to satisfy the people who complained about the first two movies being too slow and not having enough cenobite screen time. To suggest that Hellraiser II has a higher rating because it was gorier than Hellraiser III baffles me, as if III is the one with substance.