"Student painters" -- seems like a nice idea but, as in this case, they
come up with some stupid ideas.

I think the water-based stain sealed the wood and the oil-based stain
sat on top, having nowhere to go. If treatment is done consistently,
there's no reason for 2 coats. If the wood has been neglected, then
maybe a 2nd coat. But not right away, and not the same day.

Personally, I would scrape or power wash the loose stuff. The I would
wait til next year to re-treat. This will give more time for more
peeling and let the stain that's in the wood to wear. If there's not
much wear next spring, I'd wait until the fall and do 1 coat of an
oil-based stain. If there is good wear or the wood looks dry next
spring, then apply 1 coat of oil.