The first name component of the name "Boris Johnson" is Boris. It's just as valid a name as any other name he goes by, for both social and legal purposes.
His birth name, and I believe still his legal name, is Alexander Boris de Pfeffel Johnson. But he is "commonly known as" Boris Johnson, so that's what can go on the ballot paper, although his full name must still go on the registration documents. Count Binface is in the same situation.
Since he gave up his US citizenship in 2016, I don't think he's subject to a jurisdiction where "legal name" is a meaningful notion. Ultimately, this means his first name is whatever he says it is (provided that he isn't lying): you'd have to ask him. Supposing that his answer changes depending on the social context, then both "Alexander" and "Boris" would be valid answers for "what is this person's first name?".
Count Binface is in a different situation, because "Count Binface" is a stage name, which he (presumably) doesn't go as in his real life. However, stage names are still considered legal enough to go on your UK passport. Ultimately, whether he would be allowed to write "Count Binface" as his name on the registration documents would depend on what the Electoral Commission decides. I had a look last week, but couldn't find their rules.
>Ultimately, whether he would be allowed to write "Count Binface" as his name on the registration documents would depend on what the Electoral Commission decides. I had a look last week, but couldn't find their rules.
One can stand for election using any name that one is "commonly known by".
The first word in the two word phrase "Boris Johnson" is indeed Boris.
The full birth certificate name of the former "born American" PM is Alexander Boris de Pfeffel Johnson, so arguably that person's "first name component" isn't Boris alone.