There are two separate issues to address here:
1) The definition of Grooving operation vs Profiling operation.
2) Chamfering with the parting operation
I will address the second issue first, as it is easier. Adding to Support for Chamfering in the parting operations is prioritized in the backlog but not something that has active resources on it yet.
Now the first issue has to do with a fundamental understanding of how the two different turning strategies work in general, they are driven by the type of tool you are using, not the cutting direction. With that said, a grooving tool is typically used perpendicular to the spindle and a profiling tool is typically used parallel to the spindle.
I will start by showing the UI from Fusion and from HSMWorks and you can see both tool types can work parallel or perpendicular to the Axis.
Defualt Profiling definition:
Profiling tool - Parrallel to the Z Axis

Face Profiling
Profiling tool - Face profiling or -90 pass dirrection

Default Grooving opperation
Grooving tool - Perpendicular to the Z axis

Sidways grooving
Grooving tool - Sideways Grooving pattern

To throw one more curve ball, you can also you “Pass direction to set an arbitrary direction also:
45 deg pass dirrection

Or, specify a tool Orientation + a Pass Dirrection

I hope this helps
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AL Whatmough
Director Product Management - Manufacturing
Note, I love to engage on the forums. However, I spend a lot of time in meetings trying to help clear the path for our amazing team of Developers working on Manufacturing at Autodesk. So, if I don't respond immediately, it's not that I don't care.