Whats the correct way to use a ships name?

[The Wormhole]

And holo-Barclay in Projections. Multiple times. But perhaps he doesn't qualify as 'someone' saying it.

- " Oh, boy. The Voyager. Oh, boy. This is going to sound a little crazy from your perspective, but you've got to trust me. None of this is happening. This is all a holographic simulation that you've been running."
- "BARCLAY: Well, those are good questions. First, I can tell you how to destroy the Voyager. Second, if you don't do it, you're going to die. And third. Well, I'm just not an alien and I don't know how else to convince you of that."
- "You don't have memory circuits, you have a mind, and it's being damaged. Destroy the Voyager, the programme will end, and we can get you off the holodeck."

But I agree that it just sounded ... off.

In that case, I assumed they were intentionally trying to sound a bit off to fit with the reality bending theme of the episode.

 

[Shamrock Holmes]

I read an article somewhere that suggested that the reason that Voyager and Discovery are referred without the traditional definitive article (and Enterprise is increasingly though inconsistently so) is at least partly because the ship shares the name with the show.

IRL, I believe it is standard practice in the US to use "the" as the full form is "the United States Ship X", possibly partly in opposition to British/Commonweath practice to not do so as the full form there is "His* Majesties Ship Y".

*Her's from mid-20th to early 2022 and for much of the 19th obviously.