Typically what I do is separate the field used for assigning the task from the field displayed to the user.

For example, I'll include a field on the first form that tracks who it will be assigned to, then on the approver's form I have a different field that has the name of the current user as the default value.

The fields cannot be updated based on assignment, but because the "visible" field is not populated in the first step it is technically blank and will show the name of whoever opens the task.

Once that user submits the form, the name will be saved so it will show that person's name. The original field will still show who it was originally assigned to, but you only really need it on the initial form because the second field will actually track the approver moving forward.