Ooh this was creepy, I loved it.
So from what I gather, essentially all the creatures are named based on what The Speaking Friend (or an older entity if it exists) named their corpses.
1. My Still Friend:
This one is clearly a corpse. My Sleeping Friend asks them to make its noise once more, the death-rattle we later hear from The Frank in ending 3.
2. My Sleeping Friend:
While Jubilant and happy to be, their name tells the dark story of them being a corpse essentially before their transformation. Likely killed by The Speaking Friend. This one's name is probably the most hidden as they're only ever introduced to the player as already part of the "friends."
3. My Speaking Friend:
This one seems to be the oldest, and the progenitor of the situation. Its name likely comes from the fact it can talk to humans and read their language.
3. The Frank:
This one is well-spelled out in ending 3, and is essentially the name written on the poor victim's suit. We watch this man die, seemingly without any transformation (though it's implied that may come later after the story ends.)
4. My True Love:
Not sure the exact reason for the name barring it being a woman. Perhaps because My Speaking Friend specifically made them to be My Sleeping Friend's companion at request for more "friends." Regardless, her situation explains to us how this all works. In endings 1 and 2, we see the difference in mindset that can be made with My Speaking Friend's "magic." In one ending, she is horrified by her mutilated and mutated form, clinging to her previous life. In the other ending, she very clearly illustrates the mindset change that happened for My Sleeping Friend.
The child-like wonder with which these monsters carry out their unspeakable acts, was a wonderfully creepy juxtaposition.