I appreciate that Willie. I don't think the practical outcome is very similar. At least the two seemed to play very differently, IME.
OD&D had no initiative system per-se. It was left to DM judgement, or potentially to importation of something like the Chainmail system.
Holmes Basic, of course, used descending Dex order, as he had to improvise a system and used the vague references to what Dexterity does in the 1974 set as his basis, because he wasn't referencing Chainmail or the article in The Strategic Review.
B/X gives the staged system Overgeeked describes, but gives conflicting info about whether to use it as the winning side runs through all the steps, then the losing side, or whether it's meant to be interspersed. The text seems to say winners do everything THEN losers do everything, but the example of combat seems to indicate an interspersed order. IIRC BECMI is clearer.
AD&D 1E has a baseline of d6 side-based, with numerous special cases. Weapon speed factors on ties, weapon length in the round when the combatants close to melee, combatants with two attacks automatically win initiative with their first attack, then go last with their second, spells have their segments in casting time added to the initiative roll, potions have delayed onset...
AD&D 2E has a baseline of d10 individual initiative, with each weapon or action having defined speed factor, and you count up from 0 or 1; lowest rolls go first. IIRC, for example, a dagger is speed factor 2, a shortsword 3, and a longsword 5. A two hander is something awkward like 8. So it's tough for a two-hander or polearm to win initiative over a dagger, for example. In 1E the two hander would always get to strike first as the combatants close to melee, and once they engaged, the dagger would win ties. Whereas in 2E each round it's a d10+2 vs. a d10+8 or so, lowest roll strikes first.