at the start of this month or in the beginning of this month

startbeginning

According to the definition of the two words above, it's correct to say in the beginning of this month, but not at the start of this month, right?

I have no idea why you think that. "In the beginning of this month" doesn't appear anywhere in that definition. There's a good reason for that - it's not grammatically correct.

In most contexts I can think of, we'd use "at the start of ...".

We'll leave Colorado at the start of next month.
She'll arrive in Spain at the start of this month.
We arrived in Texas at the start of the month.

Some people might use "at the beginning of" in those examples but no one would use "in the beginning of" in them.