When a user shares a relationship in the system with the user who is listed as an Emergency Contact, the phrase Has Relationship will appear, indicating that these users have a relationship. When a user is added as an emergency contact but they are not in the system, then the relationship that is manually typed in is what would display. This is also why you may see differences in how the relationship titles display, such as Grandmother, grandmother, grandparent, etc. 

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Display Relationship type when pulling a list of Emergency Contacts