If I am party to a private conversation with a public figure, and that person either records (legally) the audio, a transcript, or if communication occurs via text, does that person have a legal right to publish the conversation if I explicitly state that I do not want it to be shared?
In general, you have complete free reign to publish what you like notwithstanding the wishes of the other person.
you are in a fiduciary relationship e.g. lawyer-client, doctor-patient, banker-client etc. although such obligations are ‘one-way’ e.g. the lawyer can’t but the client can.
you are bound by a non-disclosure agreement
it’s confidential information given in confidence where you have a duty of confidence. Asking someone to keep it secret only creates the second of these, the first is a function of the information the third is a function of the relationship.
disclosure would be an offence e.g. national security, operational law enforcement, contempt etc.
the person had a “reasonable expectation of privacy in all the circumstances” although this generally applies to third parties who ‘overhear’ the conversation.
When people like politicians talk “off the record” to people like journalists silence is provided by professional ethics and risk of reputation rather than legal sanctions.
does that person have a legal right to publish the conversation if I explicitly state that I do not want it to be shared?
No. The publisher would be in breach of the contract and/or breach of fiduciary duty.
Viewed as a contract, the exchange of considerations at issue was your consent to be recorded subject to your explicit request that the other person refrain from making the records publicly available (which I gather is what you mean by publish). Keep in mind, though, that a blanket prohibition to share the recording might be unenforceable under some circumstances or be stricken in court so as to facilitate fact-finding.
That being said, it would be difficult to determine the proper remedy for the violation other than injunctive relief ordering the withdrawal of the recording.
The dominant theme of the case law [...] is that fiduciary relationships arise where one party has the power and opportunity to take advantage of the other, because of that other's susceptibility or vulnerability.
The words in bold suggest that the disclosure must be unreasonable or disproportionate to the detriment of the person at jeopardy. In other words, pursuing a claim of breach of fiduciary duty despite the clear absence of harm (or likelihood thereof) could be considered frivolous.