I've seen definitions (such as the one here) that state
Homogeneous: All the terms have the same exponent.
but others (such as this one) claim that if the equation $a_n=\alpha_1 a_{n-1}+\alpha_2 a_{n-2}+...+\alpha_k a_{n-k}+h(n)$ is homogenous if $h(n)$ is 0. Are the two definitions the same and if not which is right?
"All the terms have the same exponent" is a poor way of expressing it: better to say "All the terms have the same total degree". But they are essentially the same: in $a_n = \alpha_1 a_{n-1} + \ldots + \alpha_k a_{n-k}$ we are considering the $\alpha$'s as constants and the $a$'s as variables. Each term involves one variable to the power $1$, so it's homogeneous of degree $1$. If you had a term $h(n)$ that doesn't involve any variables, that would be a term of degree $0$, so the equation would not be homogeneous. If you had a term such as $a_{n-1}^2$ or $a_{n-1} a_{n-2}$, that would have degree $2$, so again not homogeneous (and not linear).