When to use "onboard the ship" and "on board the ship"?

"Onboard", one word, is wrong.

"On-board" (hyphenated) is an adjective.

"The captain is on-board."

The on-board meals were delicious."

"On board" (not hyphenated) + noun is a prepositional modifier phrase as well as having other functions:

"We ate our dinner on board the yacht."

In both of your examples, the word "aboard" would have been much better.

OED

a. on board [...] has now, in common use, the meaning: On or in a ship, boat, etc.; into or on to a ship.

[The] fuller form [was] on ship-board (cf. Middle English ‘within schippe burdez’ [where "burdez" = sides]), and the construction ‘on board of the ship’, or ‘on board the ship’ (where ... ‘board’ means 'the deck').

[...]On board appears to be a later expansion (...) of aboard adv. and preposition, and this to have been taken directly from French à bord, ...in which bord = ‘ship's side’ comes contextually to be equal to ‘ship’ itself. ... On board, first appears late in the 17th cent.