What I'm currently reading discusses the notion of spaces being well-isomorphic, specifically a Banach space containing well-isomorphic copies of $\ell_1^n$ for every $n\in\mathbb{N}$. The author doesn't define this, so I gather it's an elementary definition, but I gather that it means for every $n$ we have a linear operator $T_n$ such that $\| x\| \leq \|T_nx\|\leq D\|x\|$ for some $D\in(0,\infty)$.
However, the only place I've actually found a definition stated is in Isomorphisms Between H1 Spaces, by Paul Muller, who just gives the requirement that $\|T_n^{-1}\|\|T_n\| <\infty$. The definition that the author of the paper I'm reading seems to be using is stronger, I think, since we don't have any constant in front of the first $\|x\|$ in the inequality. It seems that we also need to know that $\|T_n^{-1}\|=1$.
What is the proper definition?
Edit: I'm reading "An Introduction to the Ribe Program", by Assaf Naor. It's publicly available. The passage I'm considering is at the bottom of page 17.