Xjyuk

I am concerned entirely with the issue as it relates to property.

Let's say at Madeup Prep the cell phone policy allows for phones to be in pockets, but not out in class. Some teachers, however, ignore this, and have students use their phones to answer questions in real time (via something like TopHat).

Now let's say Alice has her phone out when she should not, and it is taken away by the teacher. It is returned sometime later. Then Alice's mother tells Alice's teacher that the phone is her property, not her daughter's, not the teacher's, that only Alice has permission to use it, and that if the teacher takes the phone away again, the mother considers it theft and (legal threat here).

Can the teacher/district take the phone anyway due to In Loco Parentis?

Note: For the record, the better solution for the district is to simply give detentions every time Alice violates the rule from now on, leading to eventual suspension and expulsion, but I'm far more interested in how much parenting authority a teacher/school can legally exert even when the actual parents are clearly and conspicuously opposed to such exertion.

Bonus Question: The phone is actually Alice's property, bought and paid for with money Alice earned at her job. Does this change the answer?

The school can confiscate a cell phone if you violated phone usage policy, because schools have broad powers to set student conduct policies. Searching the phone is a separate matter: a search requires reasonable suspicion and the search has to be narrowly related to that suspicion. As long as there is an actual policy and a violation of the policy, there seems to be no limit on confiscating phones.

School authority over children in the US was historically justified by reference to the in loco parentis doctrine since State v. Pendergrass, 19 N.C. 365, granting school "the authority necessary for preserving discipline", which is "analogous to that which belongs to parents, and the authority of the teacher is regarded as a delegation of parental authority". More contemporary rulings on the question of school authority, again in the domain of
corporal punishment,
as articulated in Ingraham v. Wright, 430 U.S. 651 find that

the concept of parental delegation has been replaced by the view --
more consonant with compulsory education laws -- that the State itself
may impose such corporal punishment as is reasonably necessary "for
the proper education of the child and for the maintenance of group
discipline".

In other words, the courts do not rely on the parental doctrine to justify school authority, instead they rely on what is reasonably necessary to achieve an end. Thus in New Jersey v. T.L.O, 469 U.S. 325, in loco parentis was rejected as a rationale for an unconstitutional search, reasoning

If school authorities are state actors for purposes of the
constitutional guarantees of freedom of expression and due process, it
is difficult to understand why they should be deemed to be exercising
parental rather than public authority when conducting searches of
their students.

Instead, the court frames the test in terms of competing interests:

Against the child's interest in privacy must be set the substantial
interest of teachers and administrators in maintaining discipline in
the classroom and on school grounds.

Since school authority to confiscate cell phones does not rest on acting according to the wishes of the parent, it is irrelevant that the parent approves of the child's actions. They may take the phone away, but it is not because of in loco parentis, it's because of necessity. It does not matter whose property it is; and it is not theft, because the confiscation was lawful.

In loco parentis is not necessarily dead, see Vernonia School Dist. 47J v. Acton, 515 U.S. 646, where compulsory drug testing was justified, finding that

the Policy was undertaken in furtherance of the government's
responsibilities, under a public school system, as guardian and tutor
of children entrusted to its care...

when the government acts as guardian and tutor the relevant question
is whether the search is one that a reasonable guardian and tutor
might undertake

It is not part of a school's remit to promulgate the drug-free life, so drug testing cannot be justified by appeal to necessity. Disciplinary matters are squarely within the scope of what is necessary for schools, so disciplinary questions don't need to rely on in loco parentis. In the Vernonia case, the court still finds that the state has limited reach to override the rights of children – parents still have much broader rights to restrict children than the state does. For the state, the matter has to reduce to a compelling state interest, whereas parental power isn't even subject to rational basis review.