My understanding is that privilege only applies to legal advice, and
thus wouldn't cover PR advise like that, but IANAL.
Your understanding is incorrect.
Would that count as privileged information,
The attorney-client privilege applies to confidential communications between an attorney and a client in the course of their professional relationship.
The authority supporting this varies. In some jurisdictions and in federal court, it arises as a matter of common law case precedents. In other jurisdictions, it is a statutorily created right. But all of them would agree that it includes all confidential communications in the course of the professional relationship. Most of the variation pertains to the extent of the crime-fraud exception (not applicable here).
Attorneys are expressly authorized by the rules of professional conduct to offer advice to their clients in the course of their professional relationship with a client that extends beyond advice that is strictly legal. Rule of Professional Conduct 2.1 (which has been adopted verbatim in almost every U.S. state) says:
In representing a client, a lawyer shall exercise independent
professional judgment and render candid advice. In rendering advice, a
lawyer may refer not only to law but to other considerations such as
moral, economic, social and political factors, that may be relevant to
the client's situation.
Then there is a very complicated part of this issue:
such that the lawyer had inadvertently waived privilege?
Generally, a lawyer's unilateral inadvertent action does not waive the privilege. The typical fact pattern is inadvertent inclusion of an opposing party in a "reply all" email meant to be only a "reply" to or "forward" to a client, or inadvertent inclusion of privileged material in a discovery response (e.g. due to failing to wipe metadata).
When another lawyer receives inadvertently disclosed privilege material there is a protocol to be followed.
The consequences in this case are less clear. The tweet probably wouldn't be admissible as evidence in court, unless the client ratified the disclosure. But once it is out there it isn't illegal for the inadvertent recipient, if the recipient is a non-lawyer, to share it further as was done in this case. This could even lead to a malpractice case against the errantly tweeting lawyer.