Carving knives are held by the fingers, allowing the thumb to apply leverage against the wood when carving toward you.
When carving toward yourself, the right thumb is used to provide leverage and must be kept below the travel of the blade.
When carving away from yourself, the right thumb helps grip the knife. The left thumb provides assistance pushing the blade.
Palm tools are held with the index finger and thumb towards the sharp end of the tool.
The remaining fingers are wrapped around the handle.
When making a cut with a palm tool, the thumbs are used to control the forward movement of the tool.
The right thumb is braced against the wood, while the left thumb is braced against the tool and the right thumb. This provides the maximum control.
Palm tools are nearly always "choked up on" to provide better control.
This picture shows how to hold the tool shown above.
Here is the proper hand position for making a cut with a palm tool. The left thumb is braced against the tool and the right thumb, so it acts as a brake. The right index finger acts as a stop; should the tool slip.
This is another method of holding a palm tool. In this position, the wrist provides the power and brake. This is good for removing chips, but less effective for carving detail.
This is how I use my carving tools; I am right-handed and so are the references.
I am not wearing a glove for clearer explanation of hand position.
I recommend you put on a carving glove before picking up a tool.