Initiative count is a timekeeper. If I swing on init 18 and you swing on init 12 there's no doubt in anyone's mind, either at the table or in the fiction, that I swung first.
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And if initiative count isn't a timekeeper then what's the point of it?
I don't think the first quoted paragraph is true in 3E, 4e or 5e. If you were to read Gygax's description in his DMG of the combat round, and ignore the half-baked initiative stuff, it wouldn't apply there either. The melee combat round is full of "swings" and back-and-forth and so on.
The initiative count dictates the order in which actions are declared, and so those who go first have a chance to dictate the action (within the constraints set by the rules). What that means in the fiction is pretty wide-open, though.
Now in 5e the amount of actual in-game time between the two swings is indeed fairly trivial - each pip represents about 0.3 of a second if not less - but it's still enough to suggest your swing came slightly less than two seconds after mine.
Nothing in the 3E, 4e or 5e rules says anything about pips of initiative representing any amount of time. There are not 20 segments, of 0.3 seconds each, in the 6 second round. This is an importation from Gygax's initiative rules that has no basis in the rules text of those systems, nor the game play that they support.
EDIT: Here's sufficient proof of the point that doesn't even need to point to the broader issue of rules text - suppose three characters, A with initiative 20, B with initiative 2, and C with initiative 1. Each of A, B and C can resolve 30' of movement on their turn before the next character's movement is resolved. So are you really saying that A - the most quickly-reacting of these characters who gets to have the most influence over the shape of events (by having their actions declared and resolved first) takes 5.4 seconds to move their 30 feet (just over 5' per second), while the low-DEX B moves 30 feet in 0.3 seconds (ie a speed of 100' per second)?
It makes no sense. Which reinforces the point that initiative is a way of sequencing the declaration and resolution of actions. It's not a measure of anything.