A syntax analyzer contains a syntax definition, which defines a grammar using context free grammar.
No. In table-driven parsers it contains a table which has been generated from the grammar and which drives the parser. In hand-written parsers, the code structure strongly reflects the grammar. In neither case would it be correct to say that the parser actually 'contains' the grammar. It parses the input, according to the grammar, somehow.
Is this basically 'the first part' of the syntax analyzer?
No. I don't know where you get 'first part' from.
So does the syntax analyzer include a syntax definition?
Only as described above.
After that
No, before that
the generated tokens by the Lexical Analyzer go into the Syntax Analyzer. The syntax analyzer then check, via CFG if the string input is valid
Correct.