Both were recognised as champion by the previous retired champion, before they came back, and both lost their first defence
I personally think that the Hart Burns lineage is a legitimate one. Hart did defeat Jack Johnson prior to the match, and while the decision was controversial, opinion is too divided for it to be called a robbery. Hart was matched against Jack Root for the vacant title, and Root who had defeated Hart previously, was the 3:1 betting favorite. It is not a perfect basis for establishing lineage, but it is good enough for my money.
But why is Maher not a legit champion? Thinking about it, it's a hard distinction to draw, other than stuff that happened later.
I would be willing to accept one of the lineages between Corbett retiring, and Corbett losing to Fitzsimmons, but none of them really put down roots.
I think the Fitz-Maher fight deserves to be considered a title fight for the title vacated by Corbett, Maher's claim is tenuous but probably as legit as some of the recent alfa ones.
Jeffries was paid to endorse Hart and Burns he later said he had no authority to say either was champ.
What do you see as the main differences? The big one seems to be that someone had established themselves clearly as champion when Jeffries came back. I'm not sure that should matter though.
I think Corbett playing silly buggers ,procrastinating whether he was retired or not.The original plan was for his stable mate Steve O Donnell to succeed him. Maher ruined that and when Fitz took out Maher it really left Corbett between the rock and the hard place.Corbett mustering what grace he could got into the ring after Maher had beaten O Donnell and proclaimed him champ.Maher said," to the divil with your title, Ill fight you for it!"
All fine and well until you realize the discussion is moot. Tom Sharkey was the legit champ as of 11-18-97, taking the title from Goddard. Jeffries got the title the next year, defeating Sharkey.
Root and Hart had reasonable claims to be up there as the top contenders, Maher too but O'Donnell less so. However if one recognize Fitz's claim, then Sharkey has a claim for when he defeated Fitz. He then drew with Maher and Choynski before losing to Jeffries. This is "bad" revisionism however as in reality everybody accepted Corbett as champ when he unretired.