The warbler seems to show the warm brown wing panel, very pale lower mandible and pale legs/feet of Blyth's Reed and Large-billed Reed Warbler.
I have no field experience with the latter so cannot quite comment further.
Besides, I doubt the two can be separated except in the hand?
You are making this more complicated than it really is.
well-demarcated dark breast patch = Black or Whiskered Tern
grey rump = Black or Whiskered Tern
tail pattern = Whiskered Tern
isolated brown spots on inner wing coverts = Whiskered Tern
dark alula but no dark carpal bar = Whiskered Tern...
For a start, it is probably not a good idea to compare undertail with uppertail, because you are then basically comparing inner webs to outer webs, which may have different pattern.
Even so, the pattern looks very different to me.
The WWT that you link to shows long dark streaks on its tail...
both of these birds are Whiskered Terns, not White-winged...
Photo "blacktern3" in the original post shows distinct, contrasting and thin dark tips to the tail feathers, which is typical of Whiskered (and unlike the examples of White-winged Tern posted here).