Proceed to Issue and Proceed to Record are two separate ways update a cap table.
Proceed to Issue allows you to issue options directly on Pulley and avoid having to deal with paperwork and signatures offline. This process is a faster and more cost-efficient method of issuing options to your employees. Learn more here.
Proceed to Record allows you to record a security that was issued off of Pulley on your cap table. This process is used when you've already completed the transaction offline (e.g., you've already signed your SAFEs or have already had employees sign for their option grants).