Is it necessary to examine inside the vagina to confirm the presence of semen, or is it sufficient to clean the vulva with water and perform istinja' only?

There is no need for such fears and whispers [waswas]. The default is that semen does not exit, and it is not legislated to search and investigate for it. Rather, it is a form of extremism in religion, and some scholars have considered it an innovation (bid'ah). Therefore, the questioner should rest assured and cease searching for these matters. If she is certain of seeing semen while awake or in a dream, then ghusl is obligatory. Otherwise, it is not. For istinja', wiping what is on the exit with a tissue or similar is sufficient, and ghusl is not obligatory unless the urine exceeds the exit.