In my situation where I'm incorporating the entire spectrum from 911 call (pre-hospital/Emergency Medical Services) through hospital admission/discharge. The Preliminary diagnosis (4033240) would apply to paramedic assessment (linked to clinical impression) as well as "Initial diagnosis" typically documented within an Emergency Department EMR systems.
Admitting diagnosis would represent ED admission as well as ED to Hospital admission.
"Final diagnosis" would be tied to the specific visit_occurrence and EHR's final/discharge diagnosis as documented by the local institution.
"Diagnosis present on admission" seem to be diagnosis (secondary?) known at the point of admission, but not necessarily the primary reason for admission which is identified by "Admitting diagnosis". That is a set of diagnosis acquired by nurse screening/triage, derived from presenting symptoms, history, and initial medical device reading. For example the condition of "Right bundle branch block on the EKG" as present upon admission, but the reason for admission was "Acute myocardial infarction". Or present diagnosis of "wrist sprain" and admitting diagnosis of "Stroke" - which caused the patient to fall and sprained his/her wrist.
So "Diagnosis present on admission" seem like a good way to represent one or more secondary diagnosis - which often is available in both pre-hospital and emergency department documentation.