Are these two definitions of the Mellin transform equivalent?

Question

I've been reading Chapters 6 and 7 of Bateman and Diamond's Analytic Number Theory for a more formal treatment of Mellin transforms than I've seen previously. However, I'm puzzled by the fact that, for a function $f: [1, \infty) \to \mathbb{R}$, Bateman and Diamond (BD) define the Mellin transform $F$ by $$ F(s) := \int_1^\infty f(t) t^{-s} ~dt \text{ (page 110, example 2)}. $$ Meanwhile, every other resource in which I've encountered the Mellin transform (Iwaniec and Kowalski, Diamond and Shurman, Wikipedia, etc.) uses what I believe to be the standard definition of the Mellin transform $\phi$ of a function $f: [0, \infty) \to \mathbb{R}$, $$ \phi(s) := \int_0^\infty f(t) t^s \frac{dt}{t}. $$ So I guess that brings me to two questions:

1. Are the two definitions, and their theories, somehow equivalent (in the sense that one can translate any theorem for BD's definition into a theorem for the standard Mellin transforms)?
2. Why does BD use this definition which no one else seems to use?

With regard to the first question, I wondered about some kind of $t \mapsto (t-1)^{-1}$ transformation, or maybe even splitting the integral $\int_0^\infty$ into its pieces $\int_0^1$ and $\int_1^\infty$ as one often does with Mellin transform arguments. But neither of those seemed fruitful; in fact, I've started to feel that the definitions might be incompatible/incomparable, which puzzles me even more.

With regard to the second question, I wondered if there might be some benefit to generalising the Mellin transform to broader classes of functions (which it seems to me that BD do by considering the theory of operators), and maybe the tradeoff is an atypical definition for real-valued functions on $[1,\infty)$. I also wondered if BD prefer their definition because it makes the statement of the Wiener-Ikehara theorem (BD theorem 7.3) feel very natural and clean.

Anyway, I would appreciate some perspective on how one should think about BD's Mellin transform, because I find myself unsure how it fits into the "standard" Mellin transform theory.