Gandalf: the Unilateral Coalition Builder

[Eruanna]

Could it be that Gandalf's ring was also invisible on his finger, and that Saruman just forgot Gandalf even had a ring?

I don't think that Saruman would forget any of the perceived slights against him.
However, thanks for reminding me that Saruman made his own ring. As you say Boromir there's no evidence that it did anything for him, but it does illustrate his immense knowledge of the Rings of Power and ringcraft in general.

I was still thinking about the question and I wonder if perhaps Saruman didn't try to take Gandalf's ring because he had no need of it?
Gandalf's ring has the power to enhance his abilities to influence the hearts and minds of others. Saruman, even without a Ring of Power, already possesses the ability to cloud minds and influence people, mainly through the use of his voice; as we see with Theoden and Treebeard for instance.