Your link from the Chinese embassy is not really relevant as it only mentions transit in one city without leaving the airport, which does not apply in your case.
However, China does allow 24-hour visa-free transit for most people even if travelling through multiple Chinese airports. The exact rules according to Timatic are
Passengers with a confirmed onward air, cruise or train
ticket to a third country within 24 hours can obtain an
entry permit on arrival. They must have documents required
for the next destination.
When passengers have obtained an entry permit, they can have
multiple transit stops within China (People's Rep.).
- This TWOV facility does not apply at Shenzhen (SZX) when
passengers are traveling to Hong Kong (SAR China) or Macao
- This TWOV facility does not apply at Fuzhou (FOC),
Huangshan (TXN), Mudanjiang (MDG), Urumqi (URC) and Yanji
This TWOV facility does not apply to nationals of Syria
transiting through Guangzhou (CAN).
This TWOV facility does not apply to nationals of
Afghanistan, Azerbaijan, Iran, Iraq, Kazakhstan,
Kyrgyzstan, Malaysia, Oman, Pakistan, Saudi Arabia, Sri
Lanka, Syria, Thailand, Turkey, Turkmenistan and Yemen
transiting through Xiamen (XMN).
This TWOV facility does not apply when
passengers arrive on a direct flight from USA, Guam or
Northern Mariana Isl. AND depart on a direct flight to USA,
Guam or Northern Mariana Isl. (e.g. JFK - PVG - GUM).
As none of these exceptions apply to you, you will be able to get an entry permit upon arrival and will hence not need to apply for a transit visa in advance.
TravelChinaGuide has some more information about how the 24-hour transit rule works.