Where can I find data to produce these sort of visualizations?
You're asking for administrative boundaries. How to get the official version depends on the country's administration. As far as I'm aware, a normalized dataset for all official administrative boundaries doesn't exist. IMHO the closest you might get to that is Natural Earth or OpenStreetMap.
This is not an easy task. I suggest you look at https://stackoverflow.com/questions/22166419/how-to-download-the-county-boundaries-of-china-in-openstreetmap, https://osm.wno-edv-service.de/boundaries/, http://www.naturalearthdata.com/downloads/10m-cultural-vectors/, https://mapzen.com/blog/who-s-on-first/, and https://mapzen.com/data/borders/ for a start.
What is the generic term for this level of zoom (states in US, counties in UK for example) to aid my search?
Once again, there is no normalized way of doing so, as each country uses a different naming scheme. See https://wiki.openstreetmap.org/wiki/Tag:boundary%3Dadministrative#admin_level
Also, be aware that you're using the (wrong) assumption that any given user in any given point will belong to one and just one administrative area (at any given "level"). This is not the case, specially for contested territories. Do not expect administrative boundaries to perfectly fit against each other.