If we are not going to include problems with counterfitting utility as being part of The Wirehead Problem, then I propose the term The Pornography Problem be used to refer to them.
I think umbrella category which incldues both types of problem is the main one, though. Unless a better term can be found, The Wirehead Problem seems pretty appropriate as an umbrella term. Counterfitting utility is very close to direct self-stimulation.
The classical proposed way of dealing with this is to make sure the agent has some knowledge and understanding of what its goals actually are. What the best way of doing that is is an open problem.
Daniel Dewey, 'Learning What to Value'
Abstract: I.J. Good's theory of an "intelligence explosion" predicts that ultraintelligent agents will undergo a process of repeated self-improvement. In the wake of such an event, how well our values are fulfilled will depend on whether these ultraintelligent agents continue to act desirably and as intended. We examine several design approaches, based on AIXI, that could be used to create ultraintelligent agents. In each case, we analyze the design conditions required for a successful, well-behaved ultraintelligent agent to be created. Our main contribution is an examination of value-learners, agents that learn a utility function from experience. We conclude that the design conditions on value-learners are in some ways less demanding than those on other design approaches.