For the purposes of this question, let's look at current rules first. Right now you can divide a spell or ability life cycle into 2 parts: the announcement part, where you, among other things, pay costs, choose targets and so on; and later (hopefully) the resolution part, where the “effect” of the spell or ability happens. I'll call them announcement and resolution.
They need not come immediately one after the other. A spell or ability with “instant speed” can respond to another and thus happen completely “in between” the announcement and resolution of another spell, in the following timeline:
- Announcement of spell 1
- Announcement of ability 2
- Resolution of ability 2
- Resolution of spell 1
That's how “non-interrupt fast effects” also worked during the start of the game.
There is only one difference to today, and why the term “stack” was used starting with 6th Edition rules vs. the older “batch”. Nowadays more spells and abilities can be added to the stack after just one spell or ability resolves. That was not possible with the batch—once both players passed and declined to add more to the batch, it would wind down completely until nothing was left, and the active player could start a new one.
Under the Alpha and Beta rules and up to before 6th edition, there was a third part of a spell or ability's life cycle: a window within which interrupts could mess with that spell or ability, giving us this life cycle:
- Announcement of spell or ability
- Interrupt window (use it before it closes)
During the interrupt window, only interrupt spells (like Counterspell) or abilities that said they are “played as an interrupt” (like on Ring of Immortals) could be played one after another.
But these interrupts weren't put into the batch. They were simply played one after another outside it, modifying the batch without being meaningfully a part of it. They were basically their own thing, in their own kind of “sequence” off to the side.
The tricky part, of course, was that interrupts are themselves spells (or abilities), so could be interrupted as well, in their own interrupt window. The other player would have an opportunity to interrupt, or pass and let interrupts resolve.
So a small "Counterspell war" in Alpha/Beta might have gone like this, with parts by the active player playing spell 1 and Counterspell B, while the other player plays Counterspell A
- Announcement of spell 1
- Interrupt window of spell 1
- Announcement of Counterspell A targeting spell 1
- Interrupt window of Counterspell A
- Announcement of Counterpsell B targeting Countrerspell A
- Interrupt window of Counterspell B: nothing played
- Resolution of Counterspell B: counters counterspell A
- (Counterspell A never resolves, because it was countered.)
- Resolution of spell 1
If you think this is needlessly complicated, you have a good point.
There was an idea behind it all though: Interrupts either added mana or responded to spells. Responding to spells was considered to need a “faster” speed than instant because it didn't just somehow counteract a spell (like a Giant Growth might counteract a Lightning Bolt without actually influencing the Lighning Bolt itself), but it actually "deleted" a spell (Counterspell) or modified it (make it green with Lifelace).
Under the Alpha/Beta rules, once a spell's interrupt window was over and it had “survived”, you knew it would resolve (where it might still “fizzle” for having no valid target). Once the interrupt window was over, the game would consider that the spell was “cast” or “successfully cast”. This was a trigger condition for some effects, like on Wooden Sphere where you could pay 1 generic mana to gain 1 life for a successfully cast green spell.
This means that if you had a Wooden Sphere and your green spell was countered, you couldn't gain life from the Sphere. Under current rules you do, because the trigger condition just happens immediatley after the spell is cast.
Sixth edition developments
The rules team was overhauling the rules continuously from one core set to the next, often introducing new systems in the expansions in between. It was realized between the development of 5th Edition and 6th Edition that cards like Counterspell could exist as instants with not many differences in behaviour.
So 6th Edition rules got a major overhaul of the timing rules and introduced the stack, plus damage on the stack for combat, to many complaints of players who saw the game continuously being dumbed down and intricancies being removed for "the dumb masses" that were supposedly the new target of Wizards.
In reality, when I look from time to time at some Magic questions here, having left the Magic playing community some 20 years ago, there have been so many other kinds of effects, card types and game modes added that streamling some timing things was the right thing to do, in my opinion.
It's better to have a common timing system like the stack that has relatively low complexity and then put the complexity into the cards, than have a complex dual system with batches for non-interrupts and just a "sequence" for interrupts.