The short answer is that it doesn't matter who signs an agreement first. In order for a contract to be legally binding, both parties must agree to a set of pre-defined terms (this is called “mutual assent”).
There is no general about which party should sign the contract first. From a business perspective, it is recommended that the supplier sign the contract first. If the buyer signs first they lose their leverage. When a buyer signs the contract first, it represents an offer to the supplier.
Legally it does not matter who signs the contract first as long as both parties agree to it. But, it may still be best to sign it second.
Signing orders are sequential lists that determine the order that parties receive and sign a contract in. They help streamline the signing process by ensuring that individuals sign agreements in the designated order, avoiding delays and confusion.
The authorised signatory can be the party themselves (e.g a person), a group of people (e.g. jointly liable tenants) or a signatory of the party where the party is a person (e.g. power of attorney) or a company (e.g. director).