My bathroom shower exhibits this peculiar behavior. When I turn the shower on, water runs through the spout (as expected). As soon as I lift the divertor up, water flows from the shower head (as expected). However, at the same time, I do see fairly decent flow of water continue to run through the sprout.

I have seen issues that water runs from sprout up to the shower head. Problem which I have laid out over here is exactly the reverse - Water runs through sprout, when it shouldn't be.

Any suggestions? Is it a diverter stem issue? Or anything else?

When you lift the diverter to make the water go up to the shower head there is a gasket that is suppose to cut off the water from going out the spout. These gaskets get bad and the only way to stop the problem is to replace the entire spout, there are different kinds of spouts so I would advise you to take off the old spout and take it with you to the store to make sure you get one of the same type.