FYI, I wrote the DfE and asked about this point. The response was:
You are right that information about Local Authority schools is currently missing from that section - apologies if that is misleading. We are currently working to produce more comprehensive content that covers all types of schools. In the meantime Directgov has an article
which covers this subject and which you may find useful .
So apparently we are expecting the new (unchanged?) LA school information?