Well, don't put the form code in newslettersubscription.php, put the code that you had in the plugin. (in fact you can leave the preRegister for now, there's no reason it shouldn't work).
Did you upload form.lib.php to the same directory with newslettersubscription.php?
At this point, maybe someone else has a better idea for you.
Now I'm getting scattered brained. hehehe Bear with me.
So I disabled the plugin entirely and moved that code to newslettersubscription.php. I then add {vb:raw mailform} to the template.
Now the error is:
Parse error: syntax error, unexpected T_DOUBLE_ARROW in /home/fpaul46/public_html/forum/newslettersubscription.php on line 46
Not sure what I'm doing wrong. This is what happens when a non-programmer tries to do this. (Though in my defense, it doesn't seem like it should be this complicated.)