Are the above distinctions more or less the correct ones?
Not really. First, the issue of "whether to settle or to continue litigation" generally is not about strategy or tactics. Instead, that is more of a potentially final decision a client is supposed to make on the basis of whether a foreseeable outcome compensates the injuries at issue. The lawyer is supposed to explain to the client the pros and cons of each option, at which point the client can make an informed decision the lawyer has to follow.
Second, the issue of "which, of several defendants, to sue" is not entirely a matter of strategy and tactics. The plaintiff might have personal reasons, unrelated to litigation strategy/tactics, for deciding not to sue one or more of the wrongdoers. The lawyer cannot reasonably impose himself on a plaintiff who knows what he wants. That being said, a dishonest or vexatious lawyer could influence and mislead the plaintiff's decision on who to sue even if the lawyer knows that suing those persons is futile or wrong.
Third, "day to day operating decisions" can be agreed upon in the contract or relation between a client and his attorney. Obviously a plaintiff who has considerable background in law and litigation has more elements to scrutinize the lawyer's litigation plan and suggest a different approach. On the other hand, a lawyer can always withdraw from the case if there is significant disagreement with his client on how to litigate the case.
And fourth, the phrase "a doctor or lawyer who treats themselves has a fool for a client" is rather the most typical attempt to intimidate or ridicule anyone whose knowledge of legal matters diminishes the "added value" of an attorney, since people's generalized knowledge of law & litigation threatens lawyers' profitable business of litigation.
Furthermore, equating a physician (or "doctor") with a lawyer is a laughable attempt to picture the latter as having a higher "status" of specialization. A physician's studies are vastly more complex because both curriculum and profession encompass chemistry, physics, biology, statistics, and so forth. Each one of such disciplines span various branches, lexicons, nomenclatures, etc., which the physician ought to handle and communicate with proficiency. By contrast, laws are written in language that anyone with intermediate levels of literacy can understand, most of laws being accessible online and for free without the need for specialized equipment to consult them or to take legal action. In fact, that accessibility to laws is indispensable for civilians' compliance therewith.
EDA: Per OP's edit ...
Is it true that the client mostly dictates strategy/priorities, while the lawyer has most of the say on tactics?
This might be a matter of ambiguous semantics, but here you appear to be conflating two different concepts: strategy and priorities.
A party's priorities are about what he ultimately wants, whereas strategies and tactics are about how to achieve those priorities. Or perhaps by priorities you mean the sequence of steps toward securing evidence for fact-finding, in which case that would certainly pertain to strategy.
Typically a lawyer has the freedom to discern the best way to achieve the client's priorities, but at all times the lawyer's strategies and tactics have to be centered on his client's priorities (or explain to the client why those priorities are unlawful or unrealistic). If the client insists on priorities that are unlawful or unrealistic, the lawyer can withdraw from the case even if only on grounds of refusing to contravene the rules of professional conduct.