Even though it was raining heavily, the children insisted on playing outside; they loved jumping in puddles.
Correct: D
The sentence begins with an introductory dependent clause ('Even though it was raining heavily'), which should be followed by a comma. The two main clauses ('the children insisted on playing outside' and 'they loved jumping in puddles') are independent clauses and are closely related. They can be correctly joined by a semicolon. Option D correctly places a comma after the introductory clause and uses a semicolon to connect the two independent clauses, avoiding a comma splice.