Use 'insist upon' to emphasize your strong desire for something that must happen. I insist upon receiving a proper apology.
Pro Tip 2/3
No Compromise
'Insist upon' indicates a firm stance where you're not willing to back down or compromise. He insisted upon including the clause in the contract.
Pro Tip 3/3
Expect Resistance
When you 'insist upon' something, you may be prepared for or expecting opposition or refusal. She insisted upon a refund, despite the store's no-return policy.