Is there a visa page on the passport of a visa-free country?

Yes, but please note that you must use the same passport for entry and exit, that is, you must use the same passport for entry and exit in China, the United States and Mexico (every country). For example, you can enter the United States with a business passport, then leave the United States with a business passport, then enter Mexico with a personal passport, and then leave Mexico with a personal passport. However, you can't enter the United States with a business passport and leave the United States with a private passport.

Answer supplement:

How can it be? I went abroad through a connecting flight in Europe. The border inspection in China never checked whether I had a Schengen visa, only whether I had a valid visa (mainly to Latin American countries and Africa). Of course, if you don't leave the airport, you don't need a transit visa to transfer from Europe. At present, some border inspections in China are also handled flexibly. No valid visa, not even a white passport. If you go to a visa-free country and have a round-trip ticket, some ports will also be released, such as Luohu Port, Pudong Port and Gongbei Port.

Your question is simple. Since you have a business passport and an American visa, just take your business passport through the border when you leave the country. If you need to show your ticket, just show it to the border inspectors from China to the United States, so that you can leave the United States legally. Whether it is legal to travel from the United States to Mexico is not a matter for the border inspectors from China.

Answer and add:

When you leave the country, you will naturally use your personal passport when you check the visa of the destination country. But there is a problem. Some border inspectors in China will detain you privately and won't let you leave the country. This happened at Kunming Port before. There are a group of tourists who plan to travel to Nepal (Nepal is landing visa), transfer through Bangkok (because the air tickets are cheap), and get Thai visas, but they don't have Nepalese visas. Kunming port thinks their travel plan is unreasonable (there are direct flights to Kathmandu bypassing Thailand), so although their procedures are complete according to regulations, they are still not allowed to pass. However, different ports have different standards. For example, at the three ports I mentioned earlier, as long as there are round-trip air tickets, white passports can be exempted from visas or released in landing visa countries, while some ports in the same situation cannot. So I'm worried that if you leave from Shanghai, the border inspection will not let you go because you don't have an American visa and want to transfer in the United States (in the United States and Canada, you need a transit visa even if you don't leave the airport, which the border inspection must know). But Pudong Port should be more civilized. You must use a business passport to transfer to the United States. There is no visa on your business passport. And you must use a personal passport with a Mexican visa to go to Mexico. You don't have to use the same passport for entry and exit, but you can change your passport. For example, both Australia and Britain recognize dual citizenship, so a person can leave Australia with an Australian passport and then enter Britain with a British passport. Both sides can take the national channel according to national treatment. But you must use the same passport when you return to China, that is to say, you must show your passport to Pudong for border inspection when you go abroad, and you must also use your passport when you come back, because of official and personal reasons. When you can't go out, you can use it for official business, and when you come back, you can use it for personal reasons.

Sleepless Journey 9527 reminds you, by the way, pay attention to the number of your American visas, because you have to go back and forth through the United States.

Answer and then add:

Stupid ... actually, I think I've explained it very clearly. Your understanding is correct, which is what I said above. You must use the same passport to enter and leave China, the United States and Mexico (every country). However, different passports cannot be used for entry and exit of the same country.