By working with support I learned that, if the host has the "New attendees using Chrome will join from the Web App" setting enabled, then when I click the meeting link it has me join via the web app. If the host has this option disabled then when I click the meeting link it has me join via the desktop app. (Also if I use a browser other than Chrome then it has me join via the desktop app, but Chrome is my default browser and I do not want to change that.)
This feature doesn't seem to be working as expected, though. In the GoToMeeting settings page it states that "Attendees who have already downloaded GoToMeeting will still join from the desktop app." My experience is different - it is loading the web app for me even though I have already downloaded GoToMeeting.
Support indicated that they would forward my report on to the development team for review, as they agree my experience is different than the documentation suggests.