People from Africa, SE Asia, etc. can request asylum in one of two ways:
1. if they can get a visa (travel, work, etc.) to enter the US, they can request asylum within a year of their arrival in the country.
2. if they can't get a visa, they can request asylum at their point of entry to the US (airport, seaport, or border crossing).

Yes, the refugee process is different from "standard" asylum seekers. Those seeking refugee status "must receive a referral to the U.S. Refugee Admissions Program (USRAP) for consideration." That process starts with an Executive Branch review and an official Presidential Determination, which "establishes the overall admissions levels and regional allocations of all refugees for the upcoming fiscal year." Then applications are prioritized based on referrals from the UN, US embassies, and certain NGOs, and then other factors.

There are a number of other criteria and steps, but that's my short-ish way of saying that someone can't simply go to the US embassy in Mexico to request asylum or refugee status.

The asylum seekers in the caravan were forced to wait on the Mexico side of the border because Immigration officials told them the port entry had reached full capacity, so the migrants were forced to wait until room cleared up (i.e., there was no asylum processing taking place while they waited in Mexico).

(Note: my understanding comes from reading uscis.gov; I am no expert when it comes to immigration law.)