We can insist that something be done.
We can insist on something being done.
She insisted that a zoo ticket be included in the contract.
She insisted on a zoo ticket being included in the contract.
The core meaning is the same.
She insisted that she have her own way in this matter.
She insisted on having her own way in this matter.
In the that-clause version, the verb is not simple indicative ("she has") but is shifted to the bare-infinitive ("she have"), to indicate that the statement does not describe a present reality but a desired reality. In the first example, "is" shifts to "be". This is a form of subjunctive or irrealis.
She was not insisting that a zoo ticket was already included in the contract; rather, she was expressing a resolute desire that the contract should include a zoo ticket. She wanted the contract to include a zoo ticket.