Is my name official?

Retina, 13 bytes

^[A-Z][a-z]+$

Try it online | Test suite (The output 0 means none of the strings matched, which is expected.)

When Retina is only provided with a single line of code, it outputs the number of times the expression matched the input string, so it will output 1 (truthy) if it matches and therefore is an official name and 0 (falsy) if it's not.

Breakdown

^       The beginning of the string
[A-Z]   One uppercase letter
[a-z]+  One or more lowercase letters
$       The end of the string

TeaScript, 12 bytes

xO`A-Z][a-z`

Abuses the O function.

Try this online

Test Suite

Explanation

The O function makes this:

x O   `A-Z][a-z`
x.O(/^[A-Z][a-z]+$/)

Then, the O function checks if the regex matches x.

Alternatively, a non-competing TeaScript 3 answer at 7 bytes:

xO/\A\a

JavaScript (ES6), 26

n=>/^[A-Z][a-z]+$/.test(n)

By: Edcsixtyfive

f=n=>/^[A-Z][a-z]+$/.test(n)

console.log=x=>O.textContent+=x+'\n'

;['Adnan','adnan','AdnaN','Adnan123','Adnan Adnan','A','Mary-Ann']
.forEach(t=>console.log(t+' '+f(t)))

<pre id=O></pre>