Ok, the notion on the table is that early RPG's contained no story gaming elements explicitly in the design. Is that a fair way of putting it?
No. You're ignoring the nature of the difference in clades in what appears to be claim of no difference; logically, tho, if there was no difference, there would be no debate.
In the Traditional games, the mechanics are about what a character can do. There are story considerations, rarely, such as the paladin's mount and alignment restrictions, but the mechanics are about what he can do.
Oh, and in your example of Paladins, the quest for mount doesn't exist in OD&D; it's an AD&D addition. The Paladin in OE Sup 1 just gets a horse as a special mount, and it's automatic if he's not taken a new one in the last 10 years.
In the serious storygames, there are not mechanics about what a character can do, but lots of mechanics about who can make the determination, and when a player can add stuff to the story without needing GM approval.
And there is a lot of space that's neither Traditional nor Storygame. Games where it's still about who can do what, but have certain mechanics that are about adding stuff to the narrative space in a GM-like way, but without needing to be the GM.
As an example...
Hard core storygame approach to a Star Trek Episode: A starts narrating a klingon ship. B doesn't want it; the mechanics determine how much ability B has to reject A's narration.
Hard core trad approach. The player wants a Klingon story - he thus directs the ship to the Klingon border, and hopes the GM gives him one. The GM wants a klingon story, he simply puts the klingons in, and the players either suck it up, or leave the table.
Hybrid Approach (Burning Wheel): the GM narrates a ship approaching, player uses a character skill to define that it's a klingon skill. GM sets a difficulty, and if the player makes the roll, the GM has to suck it up. If the roll fails, the GM can still choose to implement it, but in a way other than desired, or decide it really isn't a klingon.
Hybrid approach (Fate): The GM narrates a ship approaching. Player wants it to be klingons, so plunks down a fate point to declare it's a klingon before the GM describes what it is, or that it's klingons aboard a stolen ship if the GM has described some other empire's ship. GM has to suck it up, unless everyone else at table agrees it was a bad idea; if it was rejected, the player gets his fate point back.
Soft-trad approach: GM narrates a ship approaching; player asks if he can tell if it's klingon. GM considers whether or not it being klingon matters, and if not, may decide it is. Likewise, if it makes no sense, he just ignores the player's "input" into the story.