In his commentary on the first instance of the word "Elohim" in the Torah, Genesis 1.1, R' Samson Raphael Hirsch, keying off of the similar word אלה ("these"), says:
... the name of God אלוה could designate the One Whose might and will encompasses all these objects together in unity. The One through Whom all the plurality, by everything being related to Him, becomes one union, one whole, one world. Hence אלוה really means: The One Who is ruler, director, law-giver, judge of the world, מדת הדין. Hence the human rulers, directors, law-givers, judges of people of the little human world are also called אלהים.
So, according to R' Hirsch, the use of this term to mean "judges" refers to them in a lofty, nearly all-powerful sense, as not just judges but supreme rulers.
Commenting on Exodus 21:6, R' Hirsch refers back to his definition of "Elohim"-judges in Genesis and specifies that the judges referred to in this context (renewal of an Israelite servant's term of servitude) must be not just any judges, but judges with classical semicha-ordination from a chain of ordination stretching all the way back to Moshe Rabbeinu.
In his extended commentary on this section on Israelite servitude, R' Hirsch points out that "This is the one and only case in which the Torah orders a deprivation of freedom as punishment ..." (emphasis his) and goes on to discuss how very grave this treatment is, given how "personal freedom is regarded by [the Torah] so much as a holy treasure." He doesn't say so, but I'd suggest that given the gravity of this type of judgement, the Torah requires specifically judges who are specially qualified to impose control over people's lives.
In Exodus 22:7-8, the judicial activity is once again especially grave, in a different way. Here, the judges are requiring an accused thief to swear to his innocence. The Torah refers to the accused as "בעל הבית" - "the owner of the house," and R' Hirsch interprets this requirement to swear as a requirement that he stake his whole self and all of his possessions on the truth of his words, so help him God. Here, too, the judges are exerting a type of total control.
I infer that were less grave words for "judge" are used, the Torah either doesn't require judges who are qualified by ordination for such total authority, or it doesn't need to emphasize that aspect in that context. The "Shofetim" introduced at the beginning of the Parashat Shofetim are general-purpose judges who reside throughout Israel and judge routine cases, so it makes sense that they don't, as a body, need to ascribe to the authority that "Elohim"-judges bear.