Based on this screenshot of the review queue history:
It doesn't appear that review efforts performed outside of the queue contribute towards a player earning the applicable badge. I know that I had cast a VtC on the 'Player shows up inconsistently' question which I accessed by clicking on it from the front page. This action occurred before casting a Leave Open vote via the queue on a subsequent question.
I personally tend to not like reviewing through the queue since it feels unnatural for me. Rather, I go through the front page, read through the questions, and then respond accordingly (i.e. comment, comment with VtC, provide an answer). A lot of times, I'll read a question and then start clicking on linked questions or hop over to similar questions as well and sometimes I'll perform review actions there as well (especially if it's recent). In addition, there are plenty of times where I'll read a question from the queue and before issuing a vote I'll want to get a better look at what's already been done/said as it relates to comments so I'll click the question (which takes me out of the review queue) so I can see everything and perform my review action from there.
A similar question was asked about this some 6 years ago, but only 1 answer was provided and seemed uncertain.
Normally, I wouldn't give worry too much about this, however, when moderator elections occur a major component of how we rate and rank potential candidates is dependent upon certain review badges that they've earned. The system in place now favors those that complete review actions through the review queue and disadvantages those who may prefer to perform those actions in a different manner.
Is it feasible for the badge progress tracker to track review actions performed by users outside of the review queue? Is there a particular reason why we would prefer the badge tracker to only count actions done through the queue?