What actually determines whether I see H+ or 4G in the top panel?
It is the speed that determines what is shown in the indicator
For the conventions regarding the symbols and speed , see this - which are in line with what is mentioned in your question
Advertised throughput speeds are not the actual, real world throughput speeds experienced by the end user. See this for factors affecting the speed


Coming to your main concern regarding different speeds and variance with symbols displayed:

Situation described is India specific and regulated by Telecom Regulatory Authority of India. AFAIK, they are aware of operators delivering lesser speed than advertised and are in the process of streamlining it. Details are being skipped here being broad / opinionated


Operators are not at liberty to falsify speeds - some typical practices have been covered in comments
It is best to reach out to the carriers and or TRAI to get to the operator version of it