You can apply “close” to all five words of “閉じる、閉ざす、閉める、閉まる、閉てる (shimeteru or shimatteiru, I cannot read it ‘tateru’ )” you gave. The difference is the transitivity as you rightly say. The first three words are transitive and the latter two are intransitive verbs.
But there is a delicate difference of nuance and the usages of these five words. For instances, we don’t say 本を閉ざす. We say 本を閉じる. We don’t say 付き合いを閉じる、We say 付き合い(交流)を閉ざす. I cannot tell whether there are definite rules for the usages of ‘閉じる’ related words.
Here are some examples of the conjugations of the given five words all referring to “close” something.
閉じる: 目を閉じる、本を閉じる、帳簿を閉じる、窓を閉じる - close eyes (book, account book, window).
閉ざす: 口を閉ざす、耳を閉ざす、心を閉ざす、門を閉ざす、付き合いを閉ざす – close (mouth, ears, heart, gate, social association).
閉める: 戸（門）を閉める、店を閉める – close door (gate, shop).
閉まる: 戸(店)が閉まる – The door (store) closes.
閉てる: 雨戸を閉てる - close a shutter
閉めている: 店を閉めている – The shop is being closed.