I understand the Coordinate System transformation is saved with the service when you publish it. I have experimented and found that yes, if I do publish two services using 2 different transformations in Pro, the data is shifted in Pro and Web Maps.
However, I have not found anywhere I can actually see what transformation the service is using. A good place would be in the REST Page where you can see the coordinate system.
This would help us understand how my data is being displayed and aligned with features from other services.