The simple answer
A legal perspective
Is there a term describing a document that doesn't need to be signed in order to be valid?
A lawyer, doing a legal analysis of this issue, would focus on the legal meanings of the words document, signed, and valid. If you pay the lawyer by the hour, the answer will be at least 100-pages long. If you pay the lawyer a flat fee, the answer will likely be similar to,
The legal definition of "documents" is extremely broad beginning with the
invention of new technologies such as analogue audio recordings and
digital electronic records. The legal definition of "signed" includes
many actions other than the plain-English definition of a real person
physically writing the person's name on paper. Whether a document is
"valid" depends on the context of the document's use. There is no word
or phrase that describes what you are asking. In fact, because of the three issues, above, there is no
word or phrase, in law, that would properly describe "a document that needs to
be signed in order to be valid."
A plain-English perspective
I am unaware of a word or phrase in American English, except for highly specific contexts, that describes a document that must be signed to be valid other than, "a document that must be signed to be valid." Similarly, there is not a word or phrase for the more complex idea of a document that does not need a signature to be valid.
Form follows function
Words are tools we use to accomplish tasks.
In a world dominated by automobiles, for example, the English language has an abundance of commonly used, well-known words: car, sedan, truck, van, bus, motor coach, taxi, limousine, shuttle, and more. In the world without automobiles, but dominated by horses, there were many words that are no longer well-known to the general population: bit, curb bit, snaffle bit, bit shanks, hackamore, Liverpool bit, and more.
If a general word existed to describe documents that are valid only if signed, contemporary speakers of English would rarely need to accomplish tasks by using that word or the opposite of that word. If I am visiting Washington, D.C., however, and I need an automobile to go have dinner with the president or if I need an automobile to take me to the hospital, it is useful to have words such as ambulance, bus, taxi, and limousine to help me with the task of communicating exactly what I want.
Conclusion: No, such a word is impossible in a strictly legal context and it is unnecessary for contemporary life.
Addendum to address author's edit
The author of the question has provided an example: a bank printout that is official but explicitly states it does not need a signature. The author's example helps illustrate my answer, above, and does not change my answer.
In Anglo-American law, the fact that the document itself demonstratively originated from the bank, because of things such as the letterhead, it is "signed" by the bank. Because this is an English language forum and not a law forum, I will not fully explain 500 years of Anglo-American contract law and the law of evidence about why a court would absolutely rule that the document had been "signed" by the bank.
The bank's helpful note that it does not need to be signed is for non-lawyers who might be worried about the document's ability to persuade other non-lawyers (or non-bankers) of its veracity.