Thanks for thinking through this, Jon.  I think I made something unclear--responses inline.  In green to make reading easier:

On Tue, Oct 13, 2015 at 8:58 AM, Jon Robson <jrobson@wikimedia.org> wrote:
Jon thanks so much for writing this up and thanks Joaquin for putting it on the wiki (you beat me to it :))

I'm a little confused to the meaning of "tag impression" -  does it mean they rendered or that the user saw them?

User saw them, or, more specifically, appeared on screen.
 
If the latter...
https://www.mediawiki.org/wiki/Reading/Web/Projects/Categories_Browse#/media/File:Browse-Views_and_clickthrough.png suggests that 50% of the time a reader saw a tag (.45% of the time) (there was a tag impression) they clicked on them (0.18%). Am I misunderstanding what tag impression means?

Sorry, this graph is a little confusing because it has 2 axes.  The red (impressions) corresponds to the right axis and the blue (% click-through) is the left- axis.  So, the way to read it is that there were ~50,000 impressions a day, and the click-thru rate was .18%.  This means that for every ~1000 times a tag appeared to a user, ~2 clicks occurred.  The question of what % of pageviews resulted in an 'impression' is a really good one - I think it could be easily looked up using the pageview_hourly table.

Is this a fair summary?:
"When the tags were __visible__ to the user, 30-50% of the time they clicked through to the "category page". On this page only 40% of visits resulted in a click to an article in the list.
No--for reasons explained above. 

To me this is significant traffic... especially if made more visible.
I personally would expect normal link traffic to be higher and for this to be a new source of engagement.
Yes, it would be hard to get higher click-through than our current engagement model, but anything additive represents not only massive numbers, but an opportunity to have our users increasingly think of us as a place to learn (rather than a simple question and answer)
 

Can we clarify this on the wiki page and in this thread as I fear I'm misunderstanding something..?

Yes--will add clarity to the graph on wiki.  It might be awhile, however, before I have the brain and time to execute the query your question brought up of "what % of total pages lead to the tag being shown"?  This will just as easily be answered by the read more data you will be implementing.
 

Other questions:
* What was the number of clicks to tags per visit? (were being opening new tabs or clicking on one?)
I don't understand this question.  I know we can't track 'visits'.