There is no one word for this case because economics divides this up into multiple subproblems, one of which is the principal-agent problem, but that is not a unique answer to your question.
This is part of the broader discussion of external costs. Health is a quasi-good. If it were a pure good, it would also be a public good. Public goods are neither rivalrous nor excludable. It isn't rivalrous because my being healthy doesn't use up the capacity for you to be healthy. Likewise, using only lawful and ethical means, you cannot exclude people from being healthy. You could obviously kill them, but excluding such breaches, you cannot.
Insurance is a club. It is not rivalrous, but it is excludable. It is a form of voluntary sharing, whereas public goods only function under forced sharing. The problem requires a public solution as it is a public goods problem, but in the United States, we lack public health care so the patch Americans have used has been to create clubs.
This "club" system has some positive advantages and worked up until the advent of major medical insurance. The advantage is that it increases the liquidity to provide health care when needed and as such increased the amount of health care providers and drugs. In doing so, the nation became healthier, but the cost went up. To supply more physicians, nurses, drugs, facilities and other personnel it is necessary to move up the supply curve. The true demand curve is not met because the system is inefficient.
The disadvantage of major medical is that it passed all costs along to the insured to the advantage of both the provider and the insurance company. The prior system, capitation, passed all excess costs onto the patient so price increases were pretty tightly constrained. Under capitation, a physician could double their official price, but not receive any additional money because the patient couldn't pay it. Under major medical, the insurer paid all charges regardless of cost.
This led to the rise of usual, customary and reasonable charging systems, physician networks and so forth. Those are damage control systems and cannot contain costs. Conversely, the insurance company has a strong profit incentive to not permit universal coverage, even if the premiums are affordable. Since insurers basically receive a percentage over total costs, their profits depend upon rising health care costs. A system of "I decide, but you pay," is their most profitable model. Major medical exists today and capitation does not because of this.
As to the boyfriend-girlfriend problem, this is a different type of problem. The boy could be viewed as purchasing the possibility of sex and may also be viewed as purchasing a potential alliance. As the payer, it is in the boyfriend's self-interest for the girl to be happy with the purchase. The best choice would be to allow her to choose. This does a couple of things that improve efficiency.
First, the boy could always veto some idea that was too unacceptable. Further, this strategy provides information about the girl. If the goal is sex, an unacceptable movie might not be painful enough to say no to, but if the goal is a permanent mating bond, it may be enough information for the boy to know that it is not worth the emotional costs of watching terrible movies.
Second, the boy is purchasing a lottery. If the boy were risk averse, he would rather be paid than play, but sex throws risk aversion out the door and switches people to risk-loving behavior. Hence, the existence of unplanned pregnancies. How many people, just today, have said: "just this once without the condom?" Because the boy is now risk-loving, the boy is willing to pay a premium to able to take the risk. Likewise, a risk-averse girl who has become risk-loving due to the presence of hormones is willing to accept a premium for the risk that unfortunate consequences may ensue. If the premium is inadequate, then nothing will happen. However, if the premium is sufficient, it may result in no sex because the hormones have not overridden the risk aversion that is the normal state of affairs. If the guy doesn't create that "spark," the gifts won't matter.
I am reasonably certain I am more handsome than the late Hugh Heffner, but for some reason, tons of 18-22-year-old models are not flocking all over me.
There is the potential for inefficiencies in any gift-giving situation, but in the boyfriend-girlfriend case, it is probably very small in the usual case.
You can find whole classes of similar problems in the literature, especially for public, common-pool and club goods.