IANAL, but if you cross something out of a contract, sign it, and the counterparty also signs the contract, then the part you struck out is not part of the contract. The trick is getting the other person who's signing the contract to sign it even though you've crossed things out.

In my experience, people and organizations often purposely make their contracts over-reaching just to be on the safe side, and expect savvy customers/partners to cross those overreaching items off.