The "Red Cross" you said is short for, just like the Disabled Persons' Federation is called the Disabled Persons' Federation and the Women's Federation is called the Women's Federation.
"Red Cross" is a layman's name;
"Branch" is different from the name of the Red Cross Society at the next higher level. Not standardized, but the meaning is straightforward and concise, such as the name of the branch of the public security system;
"Charity Association" is a local charity federation, which belongs to the civil affairs department;
There are many "foundations", such as China Red Cross Foundation, China Youth Development Foundation, Soong Ching Ling Foundation, UNICEF and China Foundation for Poverty Alleviation.