"The movie has not started yet."
I guess the confusing thing to me is that English "start" refers strictly to the state change [from not-happening to happening], not to the ongoing state of happening. So the present tense is only valid for that transitory period. That doesn't seem to be the case for はじまる?
Quite the contrary, I think. In Japanese, that state of change is so short/transient that no-one can say 始まっています or the chances to use it are very rare, because one moment it hasn't started yet and the next moment it started already. That's why we're free to use 始まっています for the state of "having started" I think.
はじまりませんでした: In the past, it did not start. This statement is unrelated to the present.
はじまっていません: In the past, it hadn't started. In the present it may still start.
Since it's hard to clearly articulate, I'll give an example in English as well.
Did the apocalypse happen? [It did not happen.] vs [It hasn't happened.] [-masendeshita] vs [-teimasen]
Yeah I sort of had this question as well, because "the movie has already started" would be, "映画はもう始まっています" So it felt to me at first that to say "the movie has not already started" would just require me to "negate" the verb.
But I think that, "映画はもう始まっていません" would actually mean, "As for the movie, it is already not starting" which is kind of nonsense in that context; you'd need to invert the adverb too. Which is why まだ is used here to say, "映画はまだ始まっていません" or "As for the movie, it is still not starting / the movie hasn't started yet"
But yeah, that's just my 2 cents, feedback here'd be appreciated.
映画はまだ始めていません does not translate to "the movie has not started yet."
The difference between 始めていません and 始まっていません is that the first one is a transitive verb (始める）and the second one is a intransitive verb (始まる). This means that 始める is used when something is starting something else and 始まる is used when something starts itself.
映画はまだ始めていません would translate to "(Subject) has not started the movie yet." 映画はまだ始まっていません translates to "the movie has not started yet."