Revealing mistake: In the funeral scene where Dr. Viper Lady is videoing Logan fighting, she starts lowering the smartphone, but the "faked" image on the screen doesn't change to match her arm dropping down.
Revealing mistake: When Logan slices his abdomen open to pull out the parasite around his heart, watch the (digital) wound at the bottom of the screen right before he says the line "You're not gonna wanna watch this part!" Not only does a small line of blood on the left of the wound simply "blink" into existence instantly at the bottom of the screen right after he pulls the claw out (slip up in rendering), but the wound itself doesn't quite 100% match up with Logan's movements - when he breathes in, the wound doesn't really move properly with his abdomen as it inflates.
Answer: The healing factor was only reduced. Like you said, had it been removed he would have bled to death and every time he used his claws he would be bleeding from his hands. It is not revealed if it is at full strength at the end of the film, but it is likely it will return to full strength over time.