[RPG] Total distance per day for different travel paces

In D&D 5e, the daily rate of mounted overland travel is generally the same as on foot, because horses get tired and adventurers carry a lot of heavy equipment.

See the section Special Travel Pace in the DMG (p. 242–243). This section starts:

The rules on travel pace in the Player’s Handbook assume that a group of travelers adopts a pace that, over time, is unaffected by the individual members’ walking speeds. The difference between walking speeds can be significant during combat, but during an overland journey, the difference vanishes as travelers pause to catch their breath, the faster ones wait for the slower ones, and one traveler’s quickness is matched by another traveler’s endurance.

In the same section, the rule is:

• In 1 hour, you can move a number of miles equal to your speed divided by 10.

and then:

• For a fast pace, increase the rate of travel by one-third.
• For a slow pace, multiply the rate by two-thirds.

So an unencumbered horse with a speed of 60 could theoretically travel 6 miles in an hour at a normal pace. At a fast pace (a gallop), 8 miles per hour. That's "twice the usual distance for a fast pace", where "usual" means a creature with a speed of 30. This suggests that a riding horse with no rider, traveling alone, can cover 48 miles per day at a normal pace.

So the rule that "a mounted character can ride at a gallop for about an hour, covering twice the usual distance for a fast pace" seems to exist to allow for mounted travelers covering short distances quickly by using the mount's speed instead of "the usual pace", for up to an hour each day.

So, according to the rules, a traveler on a horse at a normal pace (3 miles per hour) will cover about 24 miles in an 8-hour day. If you make the horse gallop for an hour each day (fast pace for a horse being 8 miles per hour), that range increases to 29 miles. That's within the realm of what you would expect in real life, with a fast horse on good roads in fair weather.

Variant: Encumbrance

If you're using the encumbrance rule, a Riding Horse needs to be carrying less than 80 lbs of rider and equipment to get its full speed of 60. Loaded with between 80 and 160 lbs it has a speed of 50, and carrying between 160 and 480 lbs (its maximum carrying capacity) it has a speed of 30. A 200 lb adventurer in chainmail with a dungeoneering pack, longsword, and shield weighs in at about 325 lbs, so under this rule a horse's travel pace is usually the same as an unencumbered adventurer on foot.

I think that part you quoted only tries to highlight what kind of situations would you use Con. In 5e the difference between a save and a check is only what you get the proficiency from, so we can forgive their blurring of the lines here.

Also the sentence before the part you quote reads:

The DM might call for a Constitution check when you try to accomplish tasks like the following: (emphasis mine)

So it is left up to the DM and is not a "strict rule".

On whether to call for a check or a save:

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)

The DM should ask for a check when the character is doing something actively, this will also usually take one or more actions to accomplish. Saves on the other hand do not take up an action (not even a reaction) and are the result of external forces acting on the character.

So if you would rule that it takes action(s) to accomplish something, it is a check. If it does not, it is a save.

Example: holding your breath. Even though the character can deliberately try to hold their breath, they are not doing it actively: it would not cost them an action to do it. They could force open an underwater door (which definitely takes an action) while at the same time holding their breath. Also, the external force of the need of air forces the roll.