Can my opponent chain an on-summon trigger effect to my on-summon trigger effect?
You are allowed to activate cards like torrential tribute, as you mention in your own quote:  


  During the Summon response window, due to Fast Effect Timing, first
  the turn player must activate any compulsory effects that they
  triggered, such as "Summoning Curse" and "Summoner Monk", then their
  opponent must do the same. Then, the turn player has the opportunity
  to activate optional effects like "Torrential Tribute" and "Armageddon
  Knight"; if the turn player passes, then the opponent also has the
  opportunity to activate these effects. If at any point either player
  starts a Chain, standard Chain rules are followed.


The reason is that, even if another card is activated in response to the summon, that whole chain is treated as reacting to that summon.

Torrential tribute can however miss the timing, as DarkCygnus mentioned. However, this has to do with how the summon happened, and not with the cards that were activated in reaction to the summon afterwards.
Ultimate offering is indeed a good example of this, here chain link 1 resolving would be the last thing to happen.

Another example as given by Konami:


  Starlight Road: If you Chain “Starlight Road” to “Heavy Storm” while
  you control 2 or more Spell or Trap Cards and Special Summon “Stardust
  Dragon” you cannot activate “Torrential Tribute” afterwards. Even
  though the effect of “Heavy Storm” is negated, the activation was not,
  so the card will still resolve without effect and you will miss the
  timing to activate “Torrential Tribute.”


An example of Torrential tribute being chained to another effect after a summon. (it is listed under the older rulings from the UDE era, but those remain valid until overridden):


  "Torrential Tribute" can be activated after a Flip Summon, but it does
  not negate the Flip Effect (it is chained to the Flip Effect).