Difference between capacity and output current rating

Question

I'm linking to this example, but all battery packs i see on Ebay and Aliexpress are like this.

https://www.ebay.com/itm/4800-20000mAh-Rechargeable-Portable-Power-Li-ion-Battery-DC-12V-For-Camera-PVD/174315363467?hash=item2896015c8b:g:0SsAAOSw4tVd6gIE

Capacity is rated 20Ah yet output current is 1-2A. No where do they mention time rate. Are we to assume they mean 20Ah / 10h, that is, 2A for 10h?

Shouldn't such battery pack be able to provide more than 1-2A?? I mean, even tiny AAA can provide 0.5-1A for an hour, and this massive battery pack can do no more?

They even put discharge protection so it can definitely output only few amps, this one for example, has discharge protection set at 4.5A, yet capacity is 12.4Ah.

https://www.ebay.com/itm/3-7V-12400mAh-Rechargeable-Battery-Pack-Fishing-Lights-Dedicated-Standby-Power/323837413020?hash=item4b6636c69c:g:bS0AAOSw6fVdCJnh

What's the catch.

And what about individual batteries like this, are these really 2.2Ah or is that too false. Aren't battery packs made of batteries like this. They are. So if each can output 1 or 2 A in parallel they should give much more.

https://www.ebay.com/itm/20X-Batteria-ricaricabile-al-litio-li-ion-3-6V-3-7V-2200mAh-a-saldare-tabs/324218214429?hash=item4b7ce9581d:g:YMwAAOSwQpJe~uWl