If a boss in office induces one of his subordinates to have sex in exchange of offering a promotion and later after having sex deny to promote, will it fall under the definition of rape?
The conduct described would be illegal quid pro quo sexual harassment (whether or not the subordinate was promoted), for which a civil lawsuit by the subordinate for money damages would be authorized.
But it is not the crime of sexual assault (a.k.a. rape). This is because the subordinate consented to have sex, and an economic inducement to have sex, or fraud inducing someone to have sex, is not sufficient to invalidate this consent. Even if someone is induced to have sex by a much more severe inducement other than by force to the victim (e.g. by a threat to expose a highly confidential secret about someone's paternity or to kill someone's pet dog), this is often the crime of extortion, rather than the crime of rape (although state law would vary with regard to whether this fact pattern constitutes rape or extortion or both).
The non-performance of the deal to provide a promotion, however, would not have a civil lawsuit remedy for breach of promise or agreement because it would be a "meretricious" agreement that is void as against public policy because the consideration was for sex.
Similarly, suppose that in a place where prostitution is illegal, if a man promises a woman to pay her $500 to have sex, and she agrees in reliance upon that promisse, and he then refuses to pay her. In this case, the agreement is unenforceable in court (because it is a "meretricious" agreement that is void as against public policy), and the crime of rape has not been committed. This is true even if he never intended to pay in the first place and induced her to have sex through intentional and premeditated fraud. In that case, he would be guilty of soliciting a prostitution (even though he never intended to pay her) and she would be guilty of the crime of prostitution (even though she wasn't actually paid).
Of course, where prostitution is legal, no crime has been committed in that fact pattern and the woman has a right to sue the man for $500 for breach of contract or promissory estoppel, and to sue also (for any non-duplicative additional damages) for fraud, including punitive damages as well, if he never intended to pay her in the first place.
The only kind of fraud that obviates consent enough to constitute rape is when the person having sex is deceived about who they are having sex with (e.g. because they are wearing a blindfold and have been put in a context where they think they are having sex with someone else).
While a colorable argument could be made that the facts in the question amount to prostitution, which is illegal in almost all of the U.S., in practice, cases where the connection between the economic inducement to have sex and the sex itself are as indirect and subtle as they are in the question are never prosecuted in that manner for that offense. Prostitution cases are prosecuted, in practice, only in unambiguously clear, immediate, quid pro quo simultaneous transactional contexts between people who don't otherwise have a relationships with each other that provides context to the exchange.
Prostitution is typically such a petty offense that the prosecutorial effort needed to prove a case like this, the perception that a charge of that kind would punish a victim, and the perception that it would criminalize a large swath of socially accepted "gray area" conduct (like having sex following a date where one person buys the other dinner), all prevent prostitution laws from being prosecuted to their full theoretical extent, at least in cases not involving children (where statutory rape charges, or violation of position of trust charges, are often available anyway).
Possibly this would constitute "Rape by Fraud". E.g., quoting from the definition of rape found at https://leginfo.legislature.ca.gov/faces/codes_displaySection.xhtml?sectionNum=261.&lawCode=PEN:
(C) Was not aware, knowing, perceiving, or cognizant of the essential characteristics of the act due to the perpetrator’s fraud in fact.
(D) Was not aware, knowing, perceiving, or cognizant of the essential characteristics of the act due to the perpetrator’s fraudulent representation that the sexual penetration served a professional purpose when it served no professional purpose.
Mind you, I think it is highly unlikely that a prosecutor could win using this law (and I don't think it was really intended for this purpose).
Sex in exchange for "consideration" (something of value) is illegal in most of the United States, but not in Nevada (subject to county-size limitations). Sex with consent when with a legally competent person is not rape (this excludes minors and those mentally incapable of giving consent), so at best it would be breach of contract, except that there can be no contract since sex in exchange for something of value is illegal.
However, we are in one of the smaller counties of Nevada where it is legal. See the definitions part of the law: "“Prostitution” means engaging in sexual conduct with another person in return for a fee, monetary consideration or other thing of value". However, prostitution must be carried out in a licensed brothel (NRS 201.354: "It is unlawful for any person to engage in prostitution or solicitation therefor, except in a licensed house of prostitution").
Assuming that the arrangement does not happen as an instance of legal prostitution, then in Nevada it is at most a misdemeanor ("Engaging in prostitution or solicitation for prostitution") and most probably not going to be prosecuted. Since the arrangement would be illegal in Nevada, a breach of contract lawsuit would be dismissed.
In other states without the Nevada complication of legalized prostitution, rape is similar to how it is defined in Washington, where again consent is crucial to the definition of rape.