I consider it the same coaster for the reasons above.
However the awkward thought experiment is this - what if all the old wood that was removed for the rebuild was taken somewhere else and reassembled into the shape of the original coaster - which would then be the "original" coaster, the one on the original site or the reassembled one? The reassembled one would certainly have been considered the same coaster if there hadn't been a new one built on the original site.
A slightly more in-depth version of Trigger's broom is the Ship of Theseus:
en.m.wikipedia.org
The fact is there is no right answer to this question, it's a philosophical matter.
Anyway, since the old wood was not reassembled but instead given away in chunks and/or sold in a gift shop, I should be safe in considering it the same coaster - that's actually a shame for me since otherwise I'd have an extra cred to my count!