Many people have recognized that English vocabulary is sometimes deficient and some have tried to add words, usually without success. For example, many of the world's languages do not have gender-specific pronouns. Others, however – particularly those which have a pervasive system of grammatical gender (or have historically had such a system, as with English) – have gender-specificity in certain of their pronouns, particularly personal pronouns of the third person. Problems of usage arise in languages such as English, in contexts where a person of unspecified gender is being referred to, but the most natural available pronouns (he or she) are gender-specific. Michael Newdow tried to encourage us to substitute re, rees, and erm for gender non-neutral he, his, and him.
There is a similar, but more subtle, problem with the words belief and proof. The word belief is broad, it encompasses both justified and non-justified beliefs. The word proof is narrow, it designates an established fact. The problem is that we lack a good noun for properly justified belief that falls short of proof.
One symptom of this vocabulary deficiency is that some people overuse the word proof. If you see a link to an article with a title like "proof of existence of god" or "disproof of existence of gods" the best course of action is likely to not click on that link. Another result is that some people refuse to utilize the word belief on the grounds that they do not want to taint their well justified beliefs with the unjustified beliefs of the hoi polloi by sharing the same noun.
I do not know of any efforts to coin a new word that is synonymous with justified belief. The obvious problem is that the same hoi polloi will misuse the new word much like they already misuse the word proof. The goal of avoiding a "guilt by association" type of taint by utilizing a more obscure vocabulary also has a cost. Using a different word implies a different entity, but the entity itself is the same here. Belief is not the problem to be avoided. The problem is holding beliefs that are poorly justified.
There are some people who go so far as to claim that belief is the problem. I strongly disagree. A belief is what we have whenever the available evidences overall have a favored direction. I keep encountering people who claim that atheism is about having no beliefs. Yet I can no more not have a belief about the existence of gods than I can not have a belief about the existence of Leprechauns. The evidences do speak on these existence questions, the evidences are not neutral, and therefore the corresponding belief follows. We can expect one set of evidences if those entities existed, and a different set of evidences if they do not exist.
With an ultrasound we can detect pregnancy. That is proof. With Leprechauns, as with gods, as with the supernatural more generally, we can only infer. It is still proper to utilize the word proof in contexts where conclusions are inferred provided that there is a consensus of the experts that the conclusion is true. Without this usage restriction the word proof becomes too political and loses much of its meaning. There is a sense in which we arguably do have such a consensus - scientists de-facto abandoned supernaturalism several centuries ago. Unfortunately many scientists are unwilling to acknowledge that this constitutes a consensus answer to this question.
Evidences can speak loudly by being consistent and pervasive, which is the case here. We do not need proof to have properly justified strong beliefs. Mind is an natural emergent property of particular physical configurations of matter-energy, matter-energy is not a supernatural product of a non-physical mind - so we believe.