An independent clause is a °clause that can occur on its own in non-elliptical contexts.
Independent clauses are often called "main clauses", but this latter term is more often used for matrix clauses, i.e. in contrast to subordinate clauses which are contained in them. A sequence of independent clauses making up a paragraph is rarely called a sequence of "main clauses". ❡
Independent clauses are typically called "sentences", but a sentence is just a special kind of clause (a maximal clause).
main clause (CXN) = a clause that is pragmatically asserted, typically in the context of identifying the pragmatically asserted clause in a complex sentence construction. Example: in Jerry played the guitar while Phil played the bass, the clause Jerry played the guitar is the main clause, whereas while Phil played the bass is a subordinate clause. Main clauses are generally matrix clauses, but matrix clauses need not be main clauses, and dependent clauses may be pragmatically asserted – i.e. function as main clauses.