Can't there just be 3 TLs? 1 is the one we follow, 1 is Future Trunk's TL, and one is where Cell manages to kill trunks and take his Time Machine. Why is it that Cell would come from a fourth TL to the 3rd just to kill a Trunks?
TL3 Trunks went into the 4th to give Goku the medicine, exactly like the Mirai Trunks we all know and love (from TL2). Unlike that Trunks, TL3 Trunks returned to his timeline after getting the blueprints to blow up#17 THAT HE OBTAINED IN TL4, which he uses to destroy TL3's androids. When "the main Cell we all know" (which originates in this timeline: TL3) matures to Imperfect Form he finds out there's no androids. So he finds Trunks getting ready to go tell everyone he was succeful/help them with their androids/whatever we dunno or care, Cell kills TL3 Trunks and goes back in time. For reasons beyond all logic, this sends Cell into the "main timeline" (TL1) instead of the timeline TL3Trunks went to (TL4).
So can't there be just 3 TLs? If it's too confusing: YES. Only TLs 1,2 and 3 are essential to the plot. TL4 exists only to give an excuse for TL3 Cell to go back in time and enter the main (TL1) timeline. For some reason, Daizenshuu 7 gives more info on TL4 like the fact that there is a Cell games w/Perfect Cell in the same year as the Cell Games in TL1. This could ONLY take place if a Cell came back in time and absorbed TL4's androids.
I'm going to try explaining all 5 TLs backwards.
TL5: Cell goes back in time to TL4. THIS IS ALL WE KNOW OF THIS TL. Technically we don't know what year he even left, but it was probably 788.
TL4:
763: TL5 Cell arrives in embryonic form via Time Machine
764: TL3 Trunks kills Freeza+co, gives Goku medicine + warning of androids
766: Birth of this timeline's Trunks
767 Arrival of Androids, TL3 Trunks and TL4 Krillin find blueprints, what happens to TL4 Cell we don't know or care (but he is probably killed), TL3 Trunks returns to his timeline with blueprints, TL5 Cell assimilates TL4 androids and achieves perfect form, TL5 Cell holds Cell Games; results unknown.
TL3: Where Cell Comes From
764: Goku comes back to Earth (episode 120), kills Freeza+co, dies of heart disease
766: Birth of TL3 Trunks
767 Arrival of Androids, everybody but Gohan dies
780: Death of Gohan
784: Trunks goes back in time (into timeline 4's past)
785: Trunks returns with blueprints and destroys this timeline's Androids
788: TL3 Cell kills TL3 Trunks and goes into past (into TL1's past)
TL2: Where Mirai Trunks Comes From
764: Goku comes back to Earth (episode 120), kills Freeza+co, dies of heart disease
766: Birth of TL2 (Mirai) Trunks
767 Arrival of Androids, everybody but Gohan dies
780: Death of Gohan
784: Trunks goes back in time (into timeline 1's past)
785: Trunks destroys this timeline's Androids
788: Trunks destroys this timeline's Cell
TL1: The anime/manga "main" timeline
763: TL3 Cell arrives in embryonic form via Time Machine
764: TL2 (Mirai) Trunks kills Freeza+co, gives Goku medicine + warning of androids
766: Birth of this timeline's Trunks
767 Arrival of Androids, TL2 Trunks and TL1 Krillin find blueprints, TL1 Cell is killed while still in embryonic form, TL3 Cell assimilates TL1 androids and achieves perfect form, TL3 Cell holds Cell Games; results = Gohan wins.
Did that help?
Isn't Baby Trunks seen during the Trunks TV special prologue. Wouldn't that indicate that he was born before Goku dies?
Yes it would. Either they forgot that, or went according to the manga "Trunks the story" where we see TL2 Trunks as of age (about) 13.
Also, everyone looks as they do after the Androids turn up (Yamcha with his spikey hair, Vegeta in blue armour, Bulma with short hair), again indicating that Goku likely died at a later point than the same year Freeza arrived on Earth).
Probably a mistake on TOEI's part, or that's just when they started dressing that way for 3 years.
Last note, ignore if it confuses you:
Timeline 1 (main) is affected by TL2 (Mrai Trunks) and TL3 (Cell).
Timeline 4 (which is probably why they mention this in daizenshuu) is also affected by 2 TLs: TL3 (TL3 Trunks) and never mentioned TL5 (TL5 Cell).