No, and this would be inappropriate. Nothing should ever be in Unfinished Business that was not in the New Business section of a prior meeting's minutes.
While I agree with your general statement, I disagree with your classification of my example to the category the statement covers. Revisions to a given motion that are still within the same spirit but which are of variant detail should be classified as derivatives of the same procedural process and thus I feel they should be presented with the original wording of the motion. Otherwise there seems to be little benefit to discussing the prior motion between meetings and no formal way of revising the motion before it's initial introduction that I am aware of.
The benefit of the interim discussions is to allow people to be heard and allow more back-and-forth discussion. Some organizations hold "open hearings" outside the regular business session to allow this. As the discussions in-meeting get longer and more derailed, I am more likely to recommend shorter speaking limits.
When a motion is first presented, it comes under New Business. All amendments proposed at that meeting are also recorded under New Business, even if they were discussed prior to the meeting.
If the motion is tabled, it returns at the next meeting under Unfinished Business. Any further amendments presented at that meeting, regardless of whether they were discussed prior to the meeting, will also be recorded under Unfinished Business.
A motion can be revised prior to its introduction in discussion with the committee or small group intending to present it. In the end, however, if you are not the party presenting the motion, you do not have a right or privilege to modify the motion except by proposing an amendment.
