Attempts to achieve vs. attempts to avoid
The list of examples of things covered by Con checks is indeed confusing, because of the six things (counting food and water separately), it seems like four of them are are addressed by additional rules in the PHB that don't call for Con checks, while another one (going without sleep) is similarly covered by the optional rules in XGtE. So are Con checks really reserved just for drinking alcohol quickly?
Szega's answer is a great start to understanding the differences between checks and saves, and they begin by citing the relevant rules:
The DM calls for an ability check when a character or monster attempts
an action (other than an attack) that has a chance of failure. (PHB
174)
You don't normally decide to make a saving throw; you are forced to
make one because your character or monster is at risk of harm. (PHB
179)
Szega also makes a great point that generally saves are called for by other rules, whereas checks are typically at a DMs option. I would add that as such, checks are part of how a DM describes the environment, by distinguishing between what you can do automatically, and what has a chance of failure and thus needs to be rolled.
However, at least from my perspective, Szega gets tripped up by the word "action" in the description of checks, mistakenly assumes it means Action, and goes on to conclude that checks are called for when you have spent an Action on something. I don't think that is true, as a look at other checks will quickly demonstrate. "Walking across a tightrope", for example, calls for a Dexterity check, even though this is done as part of a character's Movement, with no Action having been spent. I think the quote about 'what checks are for' means "actions" in the standard English sense of the word - when a character is simply attempting to do something. So I would say:
Checks are for when a character is attempting to achieve something desired, of their own election. Success on a check means they get what they want, failure means they do not achieve it.
Saves are for when external conditions might impose a negative effect on a character. They are forced to make the save, and the cost of failure is the full imposition of the negative consequence.1 Success on the save is the avoidance, or partial avoidance, of the negative effect.
From this perspective, looking down the list of Con check examples, I would explain them as follows:
Holding Your Breath
If a character goes under water and elects to hold their breath, they can make a Con check - they are trying to achieve something positive. (Note, however, that the rules on Suffocating impose a limit on this - 1 + your Con mod in minutes. You can't just keep making Con checks and hold your breath indefinitely). If, however, the character is in the area of a stinking cloud spell, they are forced to make a Con save to hold their breath. They did not elect this, but they are forced to save in order to avoid something negative (in this case, losing their action to retching and reeling).
Marching or laboring for hours without rest
Suppose a character has already moved for eight hours in a day. They would like to keep marching - that is their desired goal. The DM might call for a Con check for them to exercise that election. If they fail, they might be told, 'no, you are just too tired to continue' with that as the only consequence of failure - they don't get what they want. If they succeed at the check and continue to march for more than an hour longer, though, then they are forced to make a Con save under the rules for a forced march. Now they are required to make a save to avoid the consequence of a level of exhaustion.
Going without sleep
Suppose an adventurer has already been up for a full day of fighting and marching. Their party camps and they take first watch. The DM informs them that they are feeling sleepy and will have difficulty completing their watch. They elect to attempt to stay awake anyway, and the DM calls for a Con check, with failure indicating that they accidentally fall asleep at some point. If the same character has been up for more than 24 hours, though, the DM might impose a Con save to avoid a level of exhaustion (as an optional rule in XGtE suggests).
Going without food and water
If a character is in a survival situation, and trying to ration their remaining supplies of food and water, they might attempt to resist the urge to eat or drink. In this case the DM might call for a Con check - success means they get what they want, and are able to hold out for longer without having to eat or drink supplies that they have. But like Suffocation, regardless of how much they resist the urge to eat or drink, the Starvation rules say that a character who goes without food for more than (3 + their Con mod in days) will take a level of exhaustion, and a character who does not get enough to drink is forced to make a Con save or be exhausted from Dehydration.
Quaffing ale
A character who tries to drink an entire stein of ale without stopping would need a Con check. Failure means they need to pause and take a breath before continuing to swallow, and in this sense it is much like making a check for attempting to hold their breath. There is no complementary save, though, because their really is no appropriate penalty when the environment forces you to try to drink something too fast. Drinking too much might call for a Con save against poison, but generally drinking too fast is simply not a danger most adventurers face.
Why aren't Con checks more common?
As I reflect on my own game, I would almost never call for any of the Con checks that I have described here. I believe that is their proper use according to the PHB, but I would not use them. So why not? Because in almost all of these cases, they fall under something we might call 'willpower', and at least in 5e willpower greatly overlaps with player autonomy. If the typical 5e player was told, "You feel too tired to march" and they responded, "I push through it and march anyway" they would expect to have their character do what they said. They would accept the consequences of exhaustion later, but would bristle at being told they couldn't choose to keep going before they reached the point of exhaustion. If they were told that, "You know you need to save your water but you can't resist the urge to drink more anyway" most of them would feel that that the DM was robbing them of agency. In general, the standard 5e genre assumes that players are in complete, rational control of their characters and can choose to ignore things like mundane tiredness, thirst, or fear. Telling a player they failed a Wisdom save and must run away from a magical fear effect is fine. Telling them they failed an Intimidation check and they are not permitted to tell off the guard is problematic.
This response is notably in contrast to other checks based on Str or Dex. I can't see many players objecting when told they were simply not strong enough to jump across a ravine or not dexterous enough to maintain their balance on ice based on failed checks. But tell them that a failed constitution check means they simply refuse to go outside because it is too cold and you will be accused of running their character for them. This response is also in contrast to other genres or systems, which have rules for piety, temptation, horror, corruption, and so forth. There it is understood that a player is not always in control of their character's decisions.
So, while I do believe that what I have described are the as-intended circumstances for constitution checks, I would be careful when applying them to your game. Certainly a "this is how I intend to use Con checks" disclaimer up front is advised.
1 The clear exception to this are the Chase rules, which require a Con check, not a Con save, to avoid Exhaustion. However, I would argue that Chase exhaustion is not "really" Condition exhaustion, and as such is more a measure of whether you are willing to continue running when you are winded as opposed to something that can actually kill you by imposing conditions on fails. As such, a check is indeed appropriate - but perhaps would be better applied at the start of your turn or when you next attempt to Dash, rather than at the end of your turn.
