Finances can be a tricky thing that needs to be spelt out and recorded to avoid misunderstandings. These two forms (see Appendix A of your house sitting agreement) are meant as a basic template to help create a formal record of the exchange of a security deposit between the house sitter and the home owner at the beginning and end of the house sitting assignment. The templates contain all the fields you should need to list all the parties' names and signatures.
Part of the security deposit lodgement form contains a template for the provision of a third party to hold the deposit and mediate its return at the end of the assignment. If you can find a neutral person to perform this service for you, all well and good. Solictors will perform this service for a fee (that may, however, be close to the amount involved). Usually, the deposit is paid to the home owner by the house sitter in the spirit of trust and good will that needs to be the foundation of every house sitting agreement of this type.
In the security deposit refund form there's space for the house sitter's expenses (for house maintenance and pet care costs etc) and liabilities (for payment for damages and any outstanding amounts owing for utilities etc) to be tallied up against the original deposit paid. There are another two forms to record these expenses in more detail.