How is the notion of a Lipschitz structure on a manifold defined? According to wikipedia, there is such a definition. $\:$ The candidate that I can come up with is

"an equivalence class of metrics that induce the topology and make the space locally bi-Lipschitz

to Euclidean space with its usual metric, where the equivalence relation is Lipschitz-equivalence".

However, I worry that that definition might be too restrictive.
(There is an obvious way to define a $\hspace{.02 in}\it{locally}$-Lipschitz structure on a manifold,

almost identically to how a differentiable structure is defined, but that's $\it{not}$ what I'm interested in.)
 A: There are two fundamental papers on this subject: Hyperbolic geometry and homeomorphisms by Sullivan, and Quasiconformal 4-manifolds by Donaldson and Sullivan. The follow-up paper Lipschitz and quasiconformal approximation of homeomorphism pairs is also relevant. All go well beyond just the definition, which is fairly obvious. 
Edit 1. I realized that I misread your question. The real answer is that the category you are interested in is equivalent to the locally Lipschitz (LL) one. Namely, every LL manifold M admits a locally bilipschitz embedding $i$ in the Euclidean space of some high dimension. Now, induce the path metric on M using the ambient Euclidean metric, i.e. define distance between points as infimum of Euclidean lengths of paths in $i(M)$ connecting these points. Then the resulting metric will satisfy your requirements. Conversely, if you have a metric on  n-manifold satisfying your requirements, then it defines a LL atlas via bilipschitz local maps to $R^n$. 
Edit 2. Here is a sketch how of you can glue together charts with different metrics while preserving the (global) bilipschitz (BL) structure, which seems to be one of your concerns (expressed in the comments). I will work with the path-metrics, which is natural in your case as you want your metrics to be locally BL to the Euclidean metric. Using the LL atlas that you have, you define rectifiable curves in a LL manifold $X$ as curves which are rectifiable in the charts (this is well-defined) and I will assume that the length structure is defined via such curves which is again natural. Given a length structure on a space $X$, you also have the notion of the integral $\int_c f(x)ds$ for continuous functions $f$ along rectifiable curves $c$. Now, given your charts $U_i$, you can define a (globally) Lipschitz partition of unity $\eta_i$ (using "distance functions to the boundary"- you have to adjust such functions slightly since your charts might be unbounded). Let I will use the notation $\int_c f(x) ds_i$ to denote the integral with respect to the length structure $\lambda_i$ coming from the $i$-th chart. Now, you define a length structure $\lambda$ on your space $X$ (it does not have to be a manifold) by
$$
\int_c ds= \sum_{i} \int_c \eta_i(x) ds_i. 
$$
Since your length structures were assumed to be BL-equivalent (this follows from BL equivalence of the distance functions), we conclude for each $i$, the restriction of $\lambda$ to $U_i$ is BL-equivalent to the length structure $\lambda_i$:
$$
C^{-1}\int_c ds_i\le \int_c ds\le C \int_c ds_i, \forall i. 
$$ 
Therefore, if you define the distance function $d_X$ on $X$ using the length structure $\lambda$, the restriction of $d_X$ to each $U_i$ is BL equivalent to the initial distance function on $U_i$.  
