If Jason drowns in a summer camp; and his next appearance shows (or talks about) him living in a forest, it seems fair to assume that Jason never returned home after surviving the drowning.
Therefore Jason's mother had no way of knowing that Jason survived; so she kept working under the assumption that he was dead.
Now, as to why he would choose to stay in the forest, that's something that (afaik) has never been answered. But let me have a few stabs in the dark:
- Jason suffered brain damage from the drowning, and has been closer to an animal than a human ever since.
- Jason suffered brain damage, retained his abilities, but lost his memory.
- Jason, being a disturbed kid (as evidenced by the murder-happy mother), started plotting to become a murderer then and there, and he basically Dextered himself by living in the woods away from society.
However, some out of universe answers also apply:
- Sequels have been known to contradict the original, because the original was not created with a sequel in mind. (example: Genie's cuffs in Aladdin)
- Horror villains have a tendency to die; and then (suprise!) appear to be still alive, if barely. Usually, a horror villain can't truly be killed, he can only be stalled. Especially if it's a series of repetitive movies using the same villain.
- Horror movies are not known for their deep or factually correct storylines.
Why was drowning declared the reason for Jason’s disappearance if there was no body to prove it?
There are plenty of cases where people are declared dead without a body being found. If Jason was last spotted in the water, no one has seen him since, and there is no Jason nor a body to be found; it stands to reason that he drowned and that the water carried him away.
Furthermore, even if someone was open to the idea that something else happened to Jason, he would need a solid lead on that. A gut feeling is not enough to (seemingly unnecessarily) drag out the investigation of a case and deny the parents closure about the death of their child.