You seem to be trying to separate the definition at the wrong place. You stop after "a personal set". However, if you read the sentence properly it makes two statements.
It reads this way as well: A personal set of religious attitudes, beliefs, and practices OR an institutionalized system of religious attitudes, beliefs, and practices.
This definition actually hinges on the word "religious". Religious and religion are not the same words even though you were trying to say they were earlier if I recall. So then what is "religious"?
Merriam-Webster defines religious as: relating to or manifesting faithful devotion to an acknowledged ultimate reality or deity
Therefore, to complete the whole description one can state:
"A religion is a personal set or an institutionalized system of attitudes, beliefs, and practices relating to or manifesting personal devotion to an acknowledged ultimate reality or deity."
Now, if one does not acknowledge some form of "ultimate reality" (inferring something beyond just "reality") or deity then they are not, by strict definition, religious.