Yes (for your own) and no (for mounts & devices):

Armor changes the movement speed of the one wearing the armor, as soon as it does encumber at all. So on this case, it is yes, you get a reduced movement speed on all of your movement ratings unless you have a feat that spares you of those.

However, keep in mind several classes have a good chance to be mounted on a being that can have other movements while in heavy armor (like a Chevalier or Paladin on a Pegasus or a Druid on a Roc). Unless the ride wears a fitted armor (barding), it will move with its own movement speed, only accounting you if you get too heavy. So no you don't account your armor on your mount's movement rating (but keep in mind it adds to your weight). Barding has the very same movement reduction as the table you referenced to - light barding doesn't hinder, but medium and heavy prevent fly speeds as of their own rules.

And then there are objects, like a magic carpet. These usually say if they work with overload and what their altered speed would be - again, it is not the armor worn by the operator, but its weight that could reduce the speed. Others like the Crab-spider-submarine don't state weight limits, in those cases, one might assume they operate at their top speed no manner how much is in them (but GM can also enforce a limit under Rule #0).

The answer to this is on page 196 of the PHB.

Using Different Speeds

If you have more than one speed, such as your walking speed and a flying speed, you can switch back and forth between your speeds during your move. Whenever you switch, subtract the distance you've already moved from the new speed. The result determines how much farther you can move. If the result is 0 or less, you can’t use the new speed during the current move.

So in the case of the Ancient Red Dragon, it can move up to 80ft each round, moving up to 40ft by walking or climbing or flying (or any mix of these), and moving the remainder by flying.