301 — Moved for good nearly always. It is the one that passes the old address's standing in search to the new page.
302 or 307 while something is temporarily away, so search engines keep the old address in their index.
410 — Gone, and not coming back for a page you have deliberately removed — a discontinued product, an event that is over. Search engines drop it from their results as they would a missing page, and it is more honest with visitors than bouncing them to your home page. Visitors see a short page that says so, with a link to your home page.
Still stuck? Ask the developers who wrote it. We aim to reply within one business day.
