Disclaimer: These are personal musings and none of it is official.
As you said, there are two origin stories for Splinter:
Splinter was Hamato Yoshi's pet rat and learned Ninjutsu from him. He witnessed the death of his human master by the hands of Oroku Saki/Shredder in an incident that also sprung him free. This is the origin story used by the original Mirage comics, the new IDW relaunch comics, the live action movies, and the 2003 animated series.
Splinter was Hamato Yoshi himself, and was mutated into Splinter due to exposure to mutagenic ooze. This is the origin story for the 1987 animated series, the Archie comics, and the new 2012 animated series.
The first origin story is rather dark, as it involves the violent death of Hamato Yoshi. The series that followed that origin story tended to be dark in nature as well.
On the other hand, the second origin story is a lot lighter. Something that's reflected in the series that follow it.
So my guess would be that the origin story was first changed by the writers of the 1987 animated series to make the story lighter and more acceptable to the norms of the time. Remember that this was back in the 80's. It would be almost unthinkable to air a cartoon that featured such a dark back story. Other changes included making the Turtles into wise cracking, pizza munching, color coded, cute super heroes. The villains became buffoons. Less emphasis was placed on the ninja aspect of the TMNT. And even then it was deemed a bit too violent by some people (i.e. the UK censors).