囲む and 囲う: people around a table and an enclosing fence
Compare a family gathered around a table with a field enclosed by a fence, identifying participants and the means of enclosure.
The family participating in 囲む
囲む（かこむ） means to surround. 「家族で丸い食卓を囲んだ。」 describes the family sitting around a round table. 家族で identifies the participating family, and 食卓を the object surrounded.
囲んだ presents gathering around the table, but it does not identify food eaten or the number of chairs. The people around the table are not an enclosing fence.
A means used with 囲う
囲う（かこう） means to enclose or surround. 「畑を低い柵で囲った。」 describes enclosing the field with a low fence. 畑を is the target, while 柵で identifies the means.
低い modifies the fence without giving its exact height. Enclosure does not establish that every kind of access was blocked or that there were no gaps.
Representative participants and means
Both examples concern surrounding a target. 家族で gives the participants around the table, while 柵で gives the means used around the field. The noun joined to で determines its role in these examples.
囲む is not limited to people, nor 囲う to fences. The comparison concerns these representative combinations of participants and means rather than exclusive definitions of the verbs.
What roles do the nouns before で have in 家族で丸い食卓を囲んだ。 and 畑を低い柵で囲った。?
Show the answer and why
家族で identifies the family participating around the table, while 柵で identifies the fence used to enclose the field. These roles do not restrict every use of the verbs to people or fences.