For a suitable theory $\sf T$, A choice principle over $\sf T$ is what is bi-interpretable with a selection principle over $\sf T$.

In the context of set theory $\sf ZF$, selection can be defined as a function from nonempty sets to their elements. Restricted selections are defined in a conditional manner by letting the function be partial and requiring the nonempty sets to meet certain conditions for selection from them to be defined. So, generally we may write:

$\operatorname {selective}(c) \iff \operatorname {function}(c) \land \forall x \in dom(c): x \neq \emptyset \to c(x) \in x$

$c$ itself can be the result of a function from prior parameters, i.e. we can have $c=F(x_1,..,x_n)$, this way we may say that $c$ is conditioned after $x_1,..,x_n$ and the function $F$. Formally, this is:

So, here the formula $\Omega$ qualifies the parameters ($x_1,..,x_n$) of the selective function as well as the field of selection $(y)$, i.e. the sets from which selections are made.

This conditional selection principle is to be denoted $\mathcal S^\Omega$.

Now, my point is that we can define choice principle by saying that: $\sf H$ is a choice principle over say $\sf ZF$, if and only if, we have a formula $\Omega$, such that: $$\sf ZF + H \rightleftharpoons ZF + \mathcal S^\Omega$$, and provided that $\sf ZF \not \vdash \mathcal S^\Omega$

Where "$\rightleftharpoons$" signify "bi-interpretability".

Bi-interpretability can work across languages. If we are to work within a single language, then perhaps simple theoretic equivalence is nicer, then for the case of $\sf ZF$, demand both:

. And of course we have: $\sf ZF \not \vdash \mathcal S^\Omega$

Seeing Asaf's posting "What is a Choice Principle, really", mentions what can be a counter-example to the above, like the Ordering Principle, that every set can be linearly ordered. And also one answer to it mentioned the principle $\neg\text{Con}(\text{ZFC})\to\text{AC}$, which was regarded as a form of conditional choice, let's call it Inconsistency Choice Principle.

So, my main question is:

Are there clear intuitively justified choice principles that those definitions fail to capture?

Related questions: Along the line of definition of choice principle given here:

Can the Ordering Principle be proved to be a choice principle?

Can the Inconsistency Choice Principle be proved to be a choice principle?

By the way, I'm not sure if intuitively these two principles actually qualify as choice principles even though they may be mentioned as such, if this line contradicts that, then I'd be happy to dismiss those qualifications as imprecise.

Working in the context of set theory $\sf ZF$, selection may be defined as a function from nonempty sets to their elements. Formally:

$\operatorname {selective}(c) \iff \operatorname {function}(c) \land \forall x \in dom(c): x \neq \emptyset \to c(x) \in x$

Restricted selections to be defined in a conditional manner by letting $c$ itself be the result of a function from prior parameters, i.e. we can have $c=F(x_1,..,x_n)$, this way we may say that $c$ is conditioned after $x_1,..,x_n$ and the function $F$, also we can impose conditions on the kind of sets from which selections are made. To capture that formally we write:

This conditional selection principle is to be denoted $\mathcal S^\Omega$.

Now, if we define choice principle over $\sf ZF$, in the following manner:

$\sf H$ is a choice principle if and only if we have a formula $\Omega$ such that both of the following are fulfilled:

. And provided that: $\sf ZF \not \vdash \mathcal S^\Omega$

Then can we prove (in $\sf ZF$ or some suitable extension of it) that the Ordering Principle [every set can be linearly ordered] is a choice principle?

[Addendum] I should relate my personal output on that issue, as a terminology I'd prefer to call principles like the Ordering Principle and the Small Violations of Choice principle [see Asaf's posting], as weak choice principles, meaning that they are actually fragments of a choice principle, i.e. something that is implied by a choice principle but cannot be proved without a choice principle. So, in nutshell a choice principle is what is equivalent to a selection principle, while a weak choice principle is what needs a choice principle to be proved. If "weak" choice principle seems confusing (of being a kind of a choice principle), then a more proper terminology in my opinion would be "fragment" of choice principle, or simply "pre-Choice" principle.

[Addendum] I should relate my personal output on that issue: principles like the Ordering Principle and the Small Violations of Choice principle [see Asaf's posting], which I think are spoken about in the context of what may be called as weak choice principles, meaning that they are fragments of $\sf AC$ that are not provable in $\sf ZF$. As, a terminology I'd prefer to call those pre-choice Principles. So, if they prove to be inequivalent with any selection principle, then they are proper pre-choice principles. This is to discriminate them from weak choice principles like $\sf DC, CC, etc..$ which are equivalent to selection principles, but weaker than $\sf AC$.

[Addendum] I should relate my personal output on that issue, as a terminology I'd prefer to call principles like the Ordering Principle and the Small Violations of Choice principle [see Asaf's posting], as weak choice principles, meaning that they are actually fragments of a choice principle, i.e. something that is implied by a choice principle but cannot be proved without a choice principle. So, in nutshell a choice principle is what is equivalent to a selection principle, while a weak choice principle is what needs a choice principle to be proved.

[Addendum] I should relate my personal output on that issue, as a terminology I'd prefer to call principles like the Ordering Principle and the Small Violations of Choice principle [see Asaf's posting], as weak choice principles, meaning that they are actually fragments of a choice principle, i.e. something that is implied by a choice principle but cannot be proved without a choice principle. So, in nutshell a choice principle is what is equivalent to a selection principle, while a weak choice principle is what needs a choice principle to be proved. If "weak" choice principle seems confusing (of being a kind of a choice principle), then a more proper terminology in my opinion would be "fragment" of choice principle, or simply "pre-Choice" principle.