There’s no difference between "I believe …" and "I believe that …" in examples A) & B) given by you. ‘That’ is subordinating conjunction that joins a clause to another on which it depends for its full meaning and that it may be let out or elided sometimes in some complex sentences. It can be let out in indirect speech, in that-clauses after some adjectives, some 2-word conjunctions and in some relative constructions.
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