There are two issues here, and I think the confusion arises when you're mixing them up.
There is stacking of
effects and there is stacking of
durations.
We all agree that multiple effects doesn't stack. Being weakened for a second time doesn't make you more weakened than you are already. This also means that if you're subject to ongoing 5 fire damage twice, you still only take five points of damage. The only twist here is that if you then take ongoing 10 fire damage, then this one supersedes the ongoing 5 ones, and you take ten points of damage instead.
So far so good. This covers stacking of effects, and it's pretty clear how it works. Then we move over to stacking of durations.
PHB278 ("overlapping durations") begins by trying to make things simpler - if you're subject to the same effect twice, simply use the one with the longer duration. For effects with fixed durations this works well, because in practice this is the same as "remember both effects, only that the one with a shorter duration will never have an impact".
But this friendly simplification doesn't work for effects with a variable duration (such as "save ends").
Now, page 279 continues by saying:
"Each round, at the end of your turn, you roll a
saving throw against each effect on you. Sometimes
an effect is a single condition or one type of ongoing
damage (page 278)."
If you read this to mean there is a rule saying you only save against each type of ongoing damage once, then, well, I can't help you. This creates an odd exception, making you save only once to get rid of three different ongoing 5 fire damage.
Instead, I believe this text only whips up a few examples, and isn't intended to be read quite so literally.
This is because if you need to save separately against each ongoing 5 fire damage
it means there are no exceptions and no confusion and the rules are clean and simple.
If you agree, we can then simply say you do keep track of each and every single effect that's impacting you. Let me show you how easy it becomes!
First, stacking of effects:
If you suffer from two immobilized conditions, they don't stack, but you remain immobilized until both effects go away.
If you suffer from two ongoing 5 fire damage, they don't stack, but you keep taking five fire damage until both effects go away.
Exactly the same! Now, stacking of durations:
If you suffer from two ongoing 5 fire damage, one until the end of the next turn, and the other until the end of the encounter (ouch!) they don't stack, and you take fire damage until both go away. (Yes, the PHB's "friendly" suggestion means you could forget about the one with the shorter duration - but really you can't, because a wizard could remove of the effects prematurely, and if he removed the end of encounter one, you'd still be affected by the one with the shorter duration)
If you suffer from two ongoing 5 fire damage (save ends), you will have to save against both. Their effects doesn't stack, but that doesn't mean they aren't separate effects whose durations won't need to be tracked separately.
I hope to have shown that any confusion that may have arisen is mostly due to the PHB trying to be friendly when really stringent rules language would have been much better.
And that the confusion arises from wordings in the PHB suggesting there might be strange and exotic exceptions to the general rules. If you read those two excerpts from the PHB literally, then yes, you could arrive at a complex mess of a ruling.
But if you don't read into the rules what I suspect wasn't intentional (only incompetent) the rules for stacking of effects and durations remain clean and easy.
And that really is a very convincing argument, at least for me!
