You should review the these following docs for more info for 8.2 SP2 to start with and I believe either way side by side or fresh install depends on how fast your upgrade/migration deadline/feasibility is:
incase you want to move only the packages from 712 to 822, then it is a straight forward by exporting and then importing the packages in 822. This is supported by SAG.
I don’t think we really need to 713 first and then to 822 migration:
What reason makes you to over install 713 first based on the documentation?..Did you face any issues while 712 → 822 migration for IS packages or TN db or MWS?
For IS, we simply upgraded the integration server’s service pack… did not installed any fixes after that not started the integartionm server after upgrade…
Yes, since we did not installed mandatory fixes for 713, incase we start the IS, we could face unwanted results… so after 712 to 713 over install, we did 713 to 822 side by side migration.