I have since found that this is neither a push message nor a cell broadcast. I disabled both, yet still received these. One of the two was already disabled by default, can't remember which.
Apparently by inserting the SIM card, I got an "app" called "MTS Menu". This is the app that shows the messages. Using App Master, I found that this app comes from /system/app/Stk.apk:

"STK" appears to stand for SIM Application Toolkit, basically a feature of SIM cards that enables operators to bundle "value added services", aka bundleware/crapware.
It is really stupid that Android comes not only with this feature enabled, but also makes the source of these pop-ups very difficult to trace.