I noticed an apparent conflict in the definition in literature about what is a "Ramanujan graph, which I was wondering if someone could kindly clarify.

(1) The Hoory-Linial-Wigderson review on expanders in its definition 5.11 calls a d-regular graph to be Ramanujan if the second highest (adjacency?) eigenvalue is bounded above by $2\sqrt{d-1}$.

(2) The Batson-Spielman-Srivastava paper (arxiv:0808.0163v3) on page 2 seems to say that a d-regular graph is Ramanujan if the non-zero Laplacian eigenvalues lie between $[d-2\sqrt{d-1},d + 2\sqrt{d-1}]$.

Aren't these two things different?

Like the largest eigenvalue of a d-regular Laplacian can I guess be as large as $2d$ even for Ramanujan graphs (and will definitely be so for bipartite d-regular Ramanujan) but how come BSS is asking for a stronger upper bound than that?

And why does Hoory-Linial-Wigderson not have any lower bound in their definition? Can the smallest eigenvalue of a d-regular adjacency matrix be arbitrarily low-right?

(3) Also the BSS paper in the footnote on page 3 says that the largest (Laplacian?) eigenvalue is at least $d+2\sqrt{d-1}$ and the second-smallest (Laplacian?) eigenvalue is at most $d-2\sqrt{d-1}$.

Are the above two bounds independent of what the Alon-Bopanna bound says as in the smallest possible value of the second-highest eigenvalue is $2\sqrt{d-1}$?