You load as part of any attack with a crossbow. You can draw the ammunition as part of the attack, so loading it is implied to be part of the attack as well. "Loaded" in this case means only that you have ammunition available for the attack. Per the official errata, "loading a one-handed
weapon requires a free hand."
This means that, post-errata, using rules-as-written (RAW), you cannot indefinitely fire two hand crossbows in the same turn. If you have a single attack, you can fire once with your action and once with the second crossbow as your bonus action with Crossbow Expert on the first turn of combat if they start loaded. However, you can't keep dual-wielding because you won't be able to reload with both hands full.
If you stow the second crossbow on the first turn with your free object interaction, you can fire the main crossbow twice on the second turn, since Crossbow Expert does not say the bonus hand crossbow needs to be different than the one-handed weapon that triggers its bonus attack. You may need to load it before the bonus attack using your object interaction, depending on your GM's interpretation of the feat. You can do this on each subsequent turn to fire a single hand crossbow twice per turn (or more than that, if you have Extra Attack).
You didn't request a rules-as-written answer, so I'll note that there is nothing game-breaking about allowing dual-wielding hand crossbows using Crossbow Expert.
Because of the way crossbows work, I think your GM would be perfectly justified in waiving the errata rule in this case but requiring you to get a customized crossbow and ammo belt to allow you to load it one-handed. That would be part of your house rules, though.
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
See "All Together Now" at the bottom for the summary.
You've kind of answered your own question. Assuming that you're level five, you have Extra Attack (1), those two feats (and you're using the sharpshooter bonus), and a Dexterity of 16...
Hand Crossbow is going to be better in that scenario.
49.5 damage for three hits is just better than 37 damage for two hits.
However, there are some things that skew this one way or the other.
Factoring in hit-chance
If you're not using the "bonus" from the Sharpshooter feat, that brings you down to 17 damage (Heavy) vs 19.5 damage (hand). It's still in favor of hand crossbow, but when combined with the points below, it might tip the scale.
Factoring in critical hits
Hits are more or less likely to be critical hits, depending on how likely it is for you to hit in the first place.
If you need a high roll to hit, then it's likely that many of your hits are going to be critical hits. Expected critical damage is 28 (Heavy) and 26.5 (Hand), if you're not using the sharpshooter bonus (48 heavy and 56.5 hand, with the bonus).
This is especially true for Champion Fighter's expanded critical range.
Keep in mind that the half-orc trait, brutal critical, and Savage Attacker only trigger on melee attacks.
Factoring in additional attacks.
(Ignoring higher ability scores)
When fighters get Extra Attack (2), expected sharpshooter damage goes up to about 55.5 (Heavy) and 66 (Hand), but expected non-sharpshooter damage to 25.5 (Heavy) and 26 (Hand)
And Extra Attack (3) brings us up to sharpshooter 74 (Heavy) and 82.5 (Hand), with non-sharpshooter damage at 34 (Heavy) and 32.5 (Hand).
The Heavy Crossbow is better here.
Additionally, things like Action Surge, Haste, etc. also lean towards Heavy Crossbow
Factoring in higher Ability Scores (and damage increases)
Increasing your Dexterity to 18 or 20 (or beyond) are going to lean this towards the hand crossbow by the increase of the modifier (+1 modifier makes it +1 damage better, etc.). This is equally true of anything that's going to bump your damage. Magic ammunition, a vulnerable enemy, Elemental weapon, etc. are all going to have a similar effect, wherein more attacks is better.
Finding a special Hand/Heavy Crossbow
A +3 Heavy Crossbow of Awesomeness is almost always going to be worth using over most mundane Hand Crossbows.
Limited ability to take bonus actions
If something is limiting your ability to take your bonus action (slow, needing it for something else, etc), then that makes the Heavy Crossbow strictly better.
All together now
Almost always, for most characters, the Hand Crossbow is going to be better if you have Crossbow Master and Sharpshooter.
The exceptions to look out for are getting significantly more attacks (Fighter Action Surge, Fighter Extra Attack, Haste, etc.), extremely hard to hit enemies (I'm not going to do the math to figure out exactly what die roll you need, but high), and finding a magic Heavy Crossbow.
Additional Notes: BBeast, rightly, points out that if your opponent is outside of the Hand Crossbow's maximum range (120), but within the Heavy Crossbow's maximum range (400), then the Heavy Crossbow wins by default.