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.
Best Answer
Yes, Crossbow Expert lets you attack with a crossbow more than once a turn.
There are two weapon properties at play here - Ammunition and Loading.
This factors into the crossbow conversation mostly due to the sometimes confusing bit about one handed weapons. The confusion comes from trying to apply it to two-handed weapons; it is quite specific about one-handed weapons.
The idea behind it is that you need one hand for the weapon and one for the ammunition, preventing you from holding a shield or second weapon. For two-handed weapons, the second hand involved in firing the weapon can also be used in the loading process.
This property (not Ammunition) restricts the rate of fire. This is also the property that Crossbow Expert removes:
Additional confusion enters when people try to use two hand crossbows to interact with the third part of Crossbow Expert:
The second crossbow takes up a hand and prevents reloading (per Ammunition). However, a hand crossbow itself is a one-handed weapon, which is an acceptable weapon to trigger that third clause and allow the bonus action attack. In other words, a character with Crossbow Expert and one hand crossbow can always fire twice a round (once with the Attack action, once with the bonus action), reloading between each shot (per Ammunition).
Characters with a hand crossbow and the Extra Attack feature can get even more shots with the Attack action, but only one from the third function's bonus action. Light Crossbows and Heavy Crossbows do not trigger the third function of Crossbow Expert (because they're two-handed weapons), and only get extra attacks from the Extra Attack feature.
I didn't touch on Crossbow Expert's second function, no-disadvantage-in-close, because it is not relevant to rate of fire.