Yes. I imagine this is not something many people have experience with. Although I have a site through Hostpapa, and have had great success with them, I can't guarantee that they would be best for this (phpbb). (Who knows maybe they would be great!) I have never even looked at another provider. From the page you linked, they all seem to offer comparable service. If it were my choice, I would probably have to take advice from the magic 8 ball.
Yeah, this entire matter goes rather over my head, but in general I find that change is always bad in the computer world.
If you absolutely must make a move, I can only suggest that you try to find a place and policies that allow you to continue in as close to the same manner you always have as possible (sorry for the awkward wording, I think you know what I mean).
There are only three kinds of people in the world: those who can count and those who can't.