You can attack three times.

Your first ambiguity isn't really ambiguous at all. The distinction between an Attack action and an attack is pretty clear. Heck, both of the quotes in your question make that distinction.

For reference, the Attack action is defined as:

The most common action to take in combat is the Attack action, whether you are swinging a sword, firing an arrow from a bow, or brawling with your fists.

With this action, you make one melee or ranged attack. See the “Making an Attack” section for the rules that govern attacks.

Certain features, such as the Extra Attack feature of the fighter, allow you to make more than one attack with this action.

Just like the quotes in your question, it refers to the Attack action and an attack as two entirely separate things. So "you can use a bonus action to attack" means that you can use a bonus action to make one attack, not to take the Attack action.

Your second ambiguity is ambiguous; it could mean that each attack with a one-handed weapon allows you to use a bonus action to attack or that the Attack action as a whole does. Fortunately, it doesn't really matter, because:

You can take only one bonus action on your turn, so you must choose which bonus action to use when you have more than one available.

So even if every attack you make with a one-handed weapon allows you to make an attack as a bonus action, you can still only take one bonus action and make one attack.

To address your comment, we have the preceding text from the definition of bonus actions:

You can take a bonus action only when a special ability, spell, or other feature of the game states that you can do something as a bonus action. You otherwise don’t have a bonus action to take.

You don't have a bonus action that Crossbow Expert allows you to make attacks with. Crossbow Expert gives you a bonus action consisting of a single attack. Even if it gives you that bonus action twice, (or up to 4 times for high level Fighters), you can still only take it once.

Official

Yes, it is confirmed by Crawford

@mpetruzz @mikemearls Crossbow Expert does allow a character to shoot a hand crossbow as an action and again as a bonus action.

Note that later, Crawford himself states that this is simply "how it is written" and that it is "murky"

@MLenser The feat is, without a doubt, murky. It's on my list of things to clear up in a future printing. @mikemearls

This tweet answer was later included in the Sage Advice Compendium

Does Crossbow Expert let you fire a hand crossbow and then fire it again as a bonus action? It does! Take a look at the feat’s third benefit. It says you can attack with a hand crossbow as a bonus action when you use the Attack action to attack with a one-handed weapon. A hand crossbow is a one-handed weapon, so it can, indeed, be used for both attacks, assuming you have a hand free to load the hand crossbow between the two attacks.

House-ruling

Ultimately, it's up to DM's discretion if it should be allowed or not. Mearls states

@GX_Sigma @Eoffram_Troyas @Wizards_DnD I'd rule it should specify a different weapon, not same weapon twice

However, note that Mearls is not the official rules guru for 5e - Crawford is - and the SAC is an official document. This tweet doesn't change the fact that the official answer is still: yes.

For balancing purposes, this is essentially giving an early Extra Attack to, for example, fighters that wouldn't be using their Bonus Action at 1st level anyway. It even adds the dex modifier to the second attack.