distinguish between the different Loyolas in terms of what they mean for a person's preparation for the job.

If I'm a hiring manager out East with a passing familiarity with BYU by reputation, but don't know exactly which intermountain west city/state they're located in, it would be easy for me to ascribe that reputation to BYU-I based on the way the credential was presented in the OP.

We can argue about the relative value of that reputation (I think BYU and BYU-I are both fine schools for my line of work, though getting into BYU speaks to pre-college preparation differently than admission to BYU-I), but I think BYU has a slightly higher profile than the various Loyolas to anyone not specifically familiar with those schools.