The Meaning of the Terms Isotropic and Anisotropic in the Total Variation Framework
In the Total Variation framework we define 2 flavors:

$$ \text{Isotropic TV} \; {TV}_{ {L}_{2} } \left( X \right) = \sum_{ij} \sqrt{ { \left( {D}_{h} X \right) }_{ij}^{2} + { \left( {D}_{v} X \right) }_{ij}^{2} } $$

$$ \text{Anisotropic TV} \; {TV}_{ {L}_{1} } \left( X \right) = \sum_{ij} \sqrt{ { \left( {D}_{h} X \right) }_{ij}^{2} } + \sqrt{{ \left( {D}_{v} X \right) }_{ij}^{2} } $$

Where $ {D}_{h} $ and $ {D}_{v} $ are the horizontal / vertical finite differences operators.

Quoting Isotropic and Anisotropic Total Variation Regularization in Electrical Impedance Tomography:


  It is worth noticing that referring to functional as ”anisotropic” is slightly misleading, because the weights corresponding to all coordinate directions are equal. 


The term Anisotropic is used because this flavor separate Vertical and Horizontal components as done in Anisotropic Diffusion. Hence it doesn't have rotational invariant term as the Isotropic Flavor. As you know, sum of quadratic terms is connected to Ellipse and with equal weights it is a Sphere / Circle which is Isotropic.

Great reading about the subject is Laurent Condat - Discrete Total Variation - New Definition and Minimization (DOI Address). It analyze the properties of different TV definitions regarding their spatial smoothing.