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As a fisherman should I be putting the vessel I worked on or the company that bought the fish?
The company/employer that paid you. I am guessing in your case the vessel is your company because you've worked for them. The company that bought the fish is your client, not your employer.
 
It's a weird one, I work for/with the skipper and tender, but we're paid directly by the buyer. The buyers have our bank details to pay us directly, based on the daily catch logs provided by us alongside our catch. The vessel, nor the individuals on it, is not a company by any measure, it is three individually self employed individuals working on a boat that someone owns (sometimes the skipper, sometimes someone else).

So I think you're thinking that the buyer buys the fish, pays the owner of the vessel/company, who then pays us?

It's more like the company buys the fish and pays the crew directly, kinda blurs the line between who the company is... I have letters of reference and catch logs from the boats I worked on and I have records of employment and (hopefully soon) letters of reference from the buyers I worked for...
 
It's a weird one, I work for/with the skipper and tender, but we're paid directly by the buyer. The buyers have our bank details to pay us directly, based on the daily catch logs provided by us alongside our catch. The vessel, nor the individuals on it, is not a company by any measure, it is three individually self-employed individuals working on a boat that someone owns (sometimes the skipper, sometimes someone else).

So I think you're thinking that the buyer buys the fish, pays the owner of the vessel/company, who then pays us?

It's more like the company buys the fish and pays the crew directly, kinda blurs the line between who the company is... I have letters of reference and catch logs from the boats I worked on and I have records of employment and (hopefully soon) letters of reference from the buyers I worked for...
Just to keep it simple, whoever pays you is your employer and you need documents such as, reference letter, pay stubs, and any other docs linked to your job from them (the one who pays you) to prove your experience.
I will give you an example; there are a lot of recruitment agencies that hire people to work for different companies, even though the work location is the company itself their employer would be the recruitment agency because they are getting paid by the agency, not the company. In your case, it doesn't matter where or with whom you work, your employer must be the person/company who is paying you for the job you've done.