This is an interesting question, and I would be curious if others are classifying vendors as you are doing. We do not break ours down into the Strategic, Tactical, Operational, or Commodity. We do things like identify if a vendor is a Fintech, is cloud-based, or handles NPI/PII.
Per our program, vendors can be categorically placed into 5 levels of risk. Very Low, Low, Medium, High, Very High. For both inherent and residual risk. Risk is calculated with a vendor risk assessment.
As far as criticality goes, we have Not Critical and Critical vendors. Critical vendors are vendors that we have operational dependence on. Meaning any prolonged outage, or loss of data is not as tolerable. Technically this measured with 4 questions here, but you can look to the OCC guidance on vendor management for a good framework.
To answer your question, I suppose it could be possible for a vendor to be Strategic, yet not critical, but I think that would not be rare. It depends on how you break down the categorical differences between strategic and operational.