I have a question regarding creating a BPM in my environment.
When I select Contact.Id in the process (as shown in Action 1.png) and then save the process, it later shows only Contact instead of Contact.Id (as shown in Action 2.png).

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Both are equivalent. When checking if a lookup (in your case Contact is a lookup on System Administration Object) is equal to some Id, Creatio will be checking the lookup's Id against that parameter. So both are equivalent, so I guess Creatio simplifies it when saving.
Hello,
This behavior is expected. At this level of the BPM, Contact and Contact.Id are effectively the same. The system automatically resolves the primary key (Id) for the Contact object, so explicitly specifying .Id is not required.
When the process is saved, the designer simplifies the reference and displays it as Contact, since the identifier is implicitly assumed. Functionally, there is no difference in execution or behavior between selecting Contact.Id and Contact in this context.
So the disappearance of .Id after saving does not indicate an issue and does not affect how the process works.