Manila - Baiyun (China) - Wuhan (China) - San Francisco

No, you don't need any kind of visa because your itineraries are definitely eligible for 24 hour visa free transit.

Of course, you should count your transit time from your first scheduled landing, until your final departure from China and not only the layovers. There is also a flight between your Chinese stops! This is very important detail.

So I just came from Chinese embassy to accompany a friend and was informed by consuls that I definitely need a visa for this 2-China stopovers.

FOR TRANSIT PASSENGER, THIS IS WHAT QUALIFIES FOR EXEMPTION:
2. Hold connected passenger tickets and are in transit to a third country or region by an international aircraft, ship or train via China, will stay for not more than 24 hours in China without leaving the port of entry.

emphasis on "in China WITHOUT LEAVING THE PORT OF ENTRY" - so if leaving 1st China stop to go to next China stop, then it doesnt qualify for TWOV

Chinese embassy is a wrong place to ask about these details because visa free transit scheme is not under the jurisdiction of the embassies and even more this option is under the jurisdiction of totally different ministry. Visa free transit is completely under the jurisdiction of Chinese immigration and you are allowed to have more than one stop within mainland China under the condition that your entire transit time, calculated from your first scheduled arrival, until your final scheduled departure from China is under 24 hours.

Also, you have not properly understood the statement that you have quoted. By the way, this statement is a clumsy translation from Chinese to English language and it would be applicable for 72 and 144 hour visa free transit only. In reality, you can have three stops within mainland China. Literally more than 100.000 passengers have successfully used this multi stop option and now you are trying to explain something that is totally wrong.

OK, you have made your mind, you will obtain a visa, but you will realize that some of your fellow travelers will travel without a visa and enjoying exactly this visa free transit option and by the way they will save their money and time that you will spend on visa obtaining. This is all fine and this is your choice. I respect that, but this still does not make you to be right on this subject.