In the Oxford dictionary, begin has the explanation: to start or to make something start for the first time.
An example:
He began a new magazine on post-war architecture.
In the Cambridge dictionary, however, it says
Start, but not begin, is used to talk about creating a new business.
- She started a new restaurant and it’s been going really well.
So how should I choose the right word?
Both "begin" or "start" can be used. But for a business, "start" is idiomatic.
The resources you have found can help you choose the best word (not the correct word, because often many different words are correct).
The first quote is ambiguous. It could mean "He began to read a magazine" or "He created a new magazine"
"Begin" means "start" in the context of the first quote, so "He started a magazine..." is also correct and would be better in the sense or "created a new magazin". The article tells you that start is used to mean "create a business".
