Which version of Illustrator are you using?
What exactly did you drag to which place?
What swatches are you going to exchange and use in InDesign?
As for your last question: Well, they are colour stops and sliders at the same time.
I've tried dragging EVERYTHING from this shown screen shot to the swatches panel.
I've found a "work around".
If the gradient is currently selected, and I create a "New Colour Group" both of the colours of the gradient are added as individual swatches to the new group.
COMPLETELY found that by accident, as I was actually creating a new group to clean up my swatches collection, and then test exporting to Indesign... but happened to have the gradient in question selected when I went to create the "New Colour Group".
Given that images of folders are the universal standard for the creation of folders, and that "New Colour Groups" have one characteristic of folders and many other characteristics not associated with folders and more common to groupings, I think it might be time for Adobe to rethink what kind of symbol should be used to indicate a "New Colour Group".
I'd suggest using brackets as an image... something like the square ones: [,,,] with commas in the middle to indicate separate but connected items in a GROUP, not a folder.
Hover over a "stop".
You'll clearly see the app tell you it's a "Gradient Slider", and reinitiate to call the slider a slider.
Hover some millimetres below or above the gradient gradient slider and you see it says that all around. It doesn't mark the stops at all, it marks just the slider
Click on a gradient stop and then open color panel. In color panel you will see a color proxy and hovering over it says that 'Gradient Stop Color'. Just drag and drop this color into the Swatches panel.
This will add your Gradient Sliders color into the swatches panel.
Alternatively, you can select the gradient filled object and choose to create a 'New Color Group' from swatches panel and then choose 'Selected Artwork' and 'Include Swatches for Tints'.This will pick all the colors of your gradient.
Hope this resolves your problem!