My observations have been, that in the past (until couple of years ago perhaps) Chinese authorities had real instructions to deal with just the simplest of foreigner related issues, and tried to navigate the rest by feeling the stones. This has realized in variable implementations across the country, each to their own interpretation of applicable laws, if any.
In my opinion one of the more vague laws is that which governs Chinese nationality and citizenship. Probably for a reason, given the political circumstances here.
In recent years this has changed. The central government has given local PSBs more detailed directives for many issues related to foreigners - in this case for children with potentially multiple citizenships, but also for foreigners' permanent residency for example.
I would think that the above reported change in policy is not so much a change of policy, but creation of common policy in place where previously there was none.
I faintly recall some article or news from Kunming about a year ago, noting that even here commercial education establishments have been kindly(?) requested to drop the "international" from their names.
It probably has more to do with penalizing false advertising, than restricting foreign early education, but the latter may be a factor too.
Maybe in comparison to coastal cities it was more welcoming years ago, but since then we've had some city administrators penalized as well.