Geminis dont need a card to state they can resummon them, because there is no such term. The only term there is with Gemini Monsters is "Normal Summon".
It's not like "Ultimate Offering", or if it is, it shouldn't logically.
Rulebook said:
*Normal Summon (and Tribute Summon)
This is the most common way to Summon a monster. Simply
place a Monster Card from your hand onto the field in faceup
Attack Position. All Normal Monsters, and most Effect
Monsters (unless they have a specific restriction), can be
Summoned in this way.
However, for Monsters that are Level 5 or higher, you must
Tribute at least 1 other monster from your side of the field
before the Normal Summon. This is called a "Tribute Summon".
Monsters that are Level 5 or 6 require 1 Tribute and Monsters
that are Level 7 or higher require 2 Tributes.
Now that's all it lists for a Normal Summon. Set is the next thing it covers, which you can do one or the other manually once per turn. "Ultimate Offering" proves it can do either by text stating "Normal Summon or Set".
"Double Summon" for one says you perform an additional Normal Summon this turn, which says two things logically to me. One, you are allowed to Normal Summon another monster this turn, and two, before doing so you Normal Summoned a monster.
I'm not sure, but if you're saying that you can Normal Summon or Set for the effect, then I would suppose "Gemini Summoner", having similar text, could both set a Gemini monster from the hand, or one already on the field.
There's a contradiction here.