Thanks for the question. I had briefly used the universe/weak-universe terminology in 2010 in my answer to a question on MathOverflow, [What interesting/nontrivial results in Algebraic geometry require the existence of universes?](https://mathoverflow.net/a/28913/1946) 

My thinking at that time was that $\kappa$ should be a *universe* cardinal, if $V_\kappa$ was an (uncountable) Grothendieck universe, which means that this is the same as an inaccessible cardinal. 

What we now call the *worldly* cardinals can be seen as a weak version of this, where one drops the requirement that $\kappa$ is regular. So $\kappa$ is worldly (or a weak universe cardinal in that MO answer), if $V_\kappa\models\text{ZFC}$. 

That answer also mentions a *very weak universe* concept, which is simply a transitive model of $\text{ZFC}$. 

I think I had created the Cantor's Attic entries at first at about the same time I had posted that answer, which was just a little before I had started using the *worldly* cardinal terminology, which I think is better and which I am happy to see has become comparatively established. So I think I may have created the early entries, and then switched over to the worldly terminology later.

The Grand Reflection cardinals, in contrast, were introduced by Philip Welch. This is a cardinal $\kappa$ relating truth in $V_\kappa$ for subsets of $V_\kappa$ to truth in $V$ for proper classes.

Incidentally, Cantor's Attic is a community-run affair, and knowledgeable people are welcome to help with adding to or improving entries.