past participle + ing Vs present perfect + ing

The word "playing" in those examples is actually a gerund, the direct object of "started" (as @BillJ points out, "started" is a catenative verb).

The difference in conjugation is between the simple past and present perfect tense of "started." Past tense ("if you started") implies an action that took place and completed at some point in the past.

On the other hand, present perfect tense ("if you have started") refers to a past action that continues to the present or has a connection to the present (PEG). In this instance, saying that you "have started the game" would imply that you are still playing the game.