I have read several statements along the lines of not changing the common name if this stays in use for a large part of the pre-split population - I assume that explains the exceptions you mention.But is this a ‘standard’ approach?
For example BirdLife split the egrets long ago. In their list and even in the Lynx ‘All the Birds of the World’ published 2020 the names Yellow-billed, Plumed and Intermediate are used. If English names are not standardised (which the aren’t) and not all world taxonomies rename parent species when a split is made, the the approach cannot be relied upon and is flawed. It will be interesting to see what names IOC use!
In fact Clements don’t even stick to the renaming principle you suggest (e.g. Savannah and Chirruping Nightjar, where the former name is retained).
I do not think your reasoning for the rename is correct or ‘a rule’, and that the renaming is probably just down to the ‘feeling’ or judgement of the taxonomic committee. You will probably find some debate somewhere about the differences in dictionary definitions of intermediate and medium, and why the later is more apt for the egret.
Intermediate can be defined as ‘coming between two things in time, place or character, etc.’ so could be argued to be less informative, but this is a rather dry and academic point - and I would prefer that academics don’t tinker with common language usage.
For specifically the former Intermediated Egret, this is the quote from the Clements website:
NielsEnglish names: To minimize confusion, the English name Intermediate Egret is now retired, and we adopt Medium Egret for the Asian species, which occurs widely alongside Little and Great Egrets. African brachyrhyncha has long been also known as Yellow-billed Egret, while the name Plumed Egret has achieved familiarity for Australian plumifera (which typically has longer breeding plumes than the other taxa), so these names are adopted here.