Yeah, I agree that would make sense, otherwise the timeline of events is a bit confusing... but do we have any explicit confirmation that this is what actually happened?
During III's final message, it states "you stand here that i would not die alone again / still i die afraid at your sister's command"; this seems to indicate that the second time III died, it was at Maya's command, but it doesn't necessarily imply that Maya killed III the first time around.
The Lore entry cited in the article (The Immanent - VI. Gloria Mundi) has Maya telling III "come, coward", but the entry doesn't indicate whether this happened before or after the events of the cutscene at the end of "The Message" mission; thus, the vision of Maya killing III could equally have been a flashback or have been showing current events as they happened.
As for why III was dead and in three-dimensional space at the start of the mission... yeah, I don't know, I might have missed a key bit of exposition, but as far as I can tell it's not explained. My assumption while playing through the mission was that Levaszk had somehow killed III by pulling it into 3D space, so that he could use its dark matter to feed the singularity more effectively, but I don't think that's explicitly stated either.
At any rate: To reduce ambiguity while avoiding speculation, I think it would be best to have the timeline of events described in the article as follows:
1. III dies at some point after the reactor meltdown mission (because Lodi is getting visions from III as the mission is happening), and its corpse is in 3D space (presumably because it got killed as it was pulled into 3D space... somehow). Whether Maya is responsible for this death is left unclear.
2. III's 3D corpse is discovered by the Guardian during "The Message", and Levaszk states that he is harvesting its mass for the singularity. Whether Levaszk is responsible for III's death is left unclear.
3. III is revived, causing its corpse to vanish, presumably because it re-entered higher-dimensional space.
4. A vision of Maya killing III is shown by III to Ikora and/or the Guardian. It is left unclear when the events in the vision occurred.
5. III dies again, and its corpse re-enters 3D space. It is left unclear whether Maya killed it this time, or whether it just died again because the Haul's revival effects were temporary.
Thoughts?