I suspect what they mean by "formal" is a "poder notarial", i.e., a power of attorney granted before a notary, but let's leave it at "formal" since this isn't mentioned explicitly. Perhaps I'm wrong, as I have little experience with Honduran law, but this seems to me a fairly vague wording.

"Carta-poder" usually means a pro forma letter of authorisation or limited power of attorney, i.e., one that you might buy a template for in a stationer's shop or office supplies store (at least here in Mexico you can). I would translate it as "proxy" or "letter of proxy" to differentiate it from the more formal and broader "power of attorney".

Proxies are agency agreements, the voting member being the principal and the proxy-holder, the agent.

In some ways, the proxy is much like a limited power of attorney in that it empowers the agent to act as if he or she was the principal at the relevant meeting at which the proxy is for. But unlike a power of attorney, the form of a proxy is generally much simpler and does not require, as might a power of attorney in most jurisdictions, onerous form requirements.
http://www.duhaime.org/LegalDictionary/P/Proxy.aspx

The last one, I presume, is just a plain business letter, though I don't understand the meaning of "dirigida a su representado" here. As Alba mentioned above, "representado" means the person/entity being represented, not the representative.

To you point about whether it's redundant: it depends on your viewpoint. If you believe "power of attorney" covers all powers of attorney, whether limited, general, notarized or simply witnessed by another party, then yes, it's redundant.

The question is, like powers of attorney themselves, whether to try to cover all possibilities or scenarios, no matter how redundant they may be, or whether to leave the wording as ambiguous as possible.

Personally, when translating legal documents, I am for translating as much of the original wording (and even redundancy) as possible even as I attempt to enhance the general readability of the output.