Assuming when you say you want two instances of the server to work at the same time you mean two separate sessions running on the same server.
The sentence "Beginning a transaction causes any pending transaction to be committed" refers to any pending transaction within the same session only. From http://dev.mysql.com/doc/refman/5.7/en/implicit-commit.html
The statements listed in this section (and any synonyms for them) implicitly end any transaction active in the current session, as if you had done a COMMIT before executing the statement.
So, if Session B starts a transaction before Session A is committed it will not force Session A to commit.