Should the word "proof" even be used when there are other words, such as inference, that have a similar interpretation? Perhaps, but it's worth remembering that there are may words that are used in different disciplines and have different meanings in each. A legal document may refer to an infant. A medical document may also refer to an infant. The lawyer writing the legal document probably is referring to someone under the age of majority. The doctor writing the medical report probably means someone less than a year or so of age. Completely different things in different disciplines.
Do we tell the doctor that he can't use the word infant because the lawyer already uses it to mean something else? No.
Just because another profession or discipline uses a word is not reason for us not to use it. A discipline is free to use words of its choosing as long as the definition of those words is clearly agreed upon by members of the discipline. The BCG Genealogical Standards Manual defines these terms. It may be that various members of the discipline disagree on some of the details.
I may not think the word "proof" is the best word to use, but if I want to play genealogy with others concerned about standards, I'll have to use agreed upon terms.
Just like I had to write mathematical proofs "by the rules" when I took Sets and Logic all those years ago.
And sometimes I didn't like those either (grin!).