There are several candidates meeting your requirements:

Screenshot (source: Google Play; click images for larger variants)
This seems to be a very suitable candidate. Not only matches your described requirements, but allows for multiple additional ways to be triggered: hotkeys and notification area included. Also included is a nice gallery of the screenshots you've already done (see above), plus the capability of editing/cropping screenshots and uploading them to different sites.
- Delay: Yes (see first screenshot; of course neither countdown nor the floating icon are captured)
- initiated by an icon: Yes, but also several different options
- Gratis: Yes.
- root required: No, but might be needed for some functions (not sure which).
- Up-to-dateness: Last release 2 month ago.
Same name, but note the "lowercase s".

screenshot (source: Google Play; click images for larger variants)
- Delay: Yes (see last screenshot)
- initiated by an icon: Yes, but also several different options (e.g. "shake")
- Gratis: Yes.
- root required: No, but might be needed for some functions (not sure which).
- Up-to-dateness: Last release a year ago.
As pointed out by the OP, there's also Touchshot with which Mawg finally ended up:

Touchshot (source: Google Play; click images for larger variants)
Though missing from the app's description, OP confirms it meets all the requirements – so I omit those details here. It's well rated (more than 4 stars at 300+ votes) and (at the point of this writing) very much up-to-date.
There are several other candidates out there. But those I know of are either completely outdated (not updated for much more than one year – so it's not clear if they'd still work on recent Android versions), or not available for free. Hope one of these two comes up to your expectations :)