Happy Lord's Day to you, too, brother!
I know the directory of family worship addresses the question — if you want to quote it, provide some commentary or defense on what it has to say
Are you referring to the Westminster Directory for Private (Family) Worship? If so, I don't think there is anything in there directly addressing who ought to "teach" (the definition of this word might be an issue). Rather, it states almost from the start that "the charge and office of interpreting the holy scriptures is a part of the ministerial calling, which
none (however otherwise qualified) should take upon him in any place, but he that is duly called thereunto by God and his kirk" (III, emphasis added). What it allows is that "in every family where there is any that can read, the holy scriptures should be
read ordinarily to the family; and it is commendable, that thereafter they confer, and by way of
conference make some good use of what hath been
read and heard." (Ibid.) added). I take "conference" to = "discuss."
Reading and discussing Scripture is different than reading and speaking authoritatively as to the meaning- if the reading and discussion leads to questions of interpretation, that should be referred to the minister. If the reading and discussion leads to questions of application, that is within the bounds of the head of household: "As, for example, if any sin be reproved in
the word read, use may be made thereof
to make all the family circumspect and watchful against the same; or if any judgment be threatened, or mentioned to have been inflicted, in that
portion of scripture which is read, use may be made to
make all the family fear lest the same or a worse judgment befall them, unless they beware of the sin that procured it: and, finally, if any duty be required, or comfort held forth in a promise, use may be made to
stir up themselves to employ Christ for strength to enable them for doing the commanded duty, and to
apply the offered comfort. In all which the master of the family is to have the chief hand; and any member of the family may propone a question or doubt for resolution." (Ibid.)
I personally consider catechizing or reading from commentaries/devotionals written by ordained ministers the same as seeking answers from the minister, with the caveat that this might include first asking the minister about the author of the latter if outside one's particular church.
As for private conversations, including PB, I don't think anyone considers that "teaching." I would say that, despite the overall dissimilarity, there is a similarity to family worship in that, if a discussion raises "a question or doubt for resolution," one should go to their minister and seek instruction.
And would the "rules" differ based on the situation of the church in a certain time and place?
I'm not sure what you mean by "situation": are you referring to eras (like early Church vs Reformation vs now) or various gatherings during the week (Sabbath School vs a mid-week Bible study)?
