In ArcGIS Online, the projection is defined as whatever the basemap is... Which is typically web mercator auxillary sphere. In Geocortex Essentials, is the projection defined the same way (whatever the basemap is?) and in terms of best practice, should the projection be web mercator auxillary sphere since it is a web mapping application? I would prefer it to be in NAD83
Geocortex Essentials pulls the web app's projection from a site's "Primary Service", which is in GCE is the bottom-most service in your "map services" configuration (Rest Manager > Map > Map Services). The default projection of that service becomes the projection for your web apps in that site. Any major projection, NAD 83 included, should work fine.
FYI, I recommend having your data in the projection that it's being consumed in to reduce server load i.e. if you want your app in NAD 83 make sure that your services and data are also all in NAD 83... it's not a big deal but it helps and would qualify as "best practice".