What's the exact criterion of dual submission? Does submitting two (substantially different) versions of a same paper count as dual submission?
To be specific, let me take this as an example:

Say I've done a research and a paper A was written about the research. However, later I found that I can make some improvements on the method of A, and after making these improvements I've got B, a second version of A. A and B are same in their abstracts, introductions and related works, but largely different in their methods and experiments. 
I've submitted B to conference X which will inform its final decision to authors in late June and I'm going to submit A to another conference Y whose submission deadline is in late April. 

Question is, will this be considered as a dual submission? If the answer is positive, then which part(s) of the content of A should I modify to avoid such a problem? Thanks.

Note that my field is Computer Science, and most conferences in this field consider their proceedings as publications.
It is up to the editor/conference committee/reviewers to decide if two papers have too much overlap.  Show them about both versions of the paper, and let them decide.

Since there is no objective way to decide if two papers are adequately different, the criterion is the opinion of the editor.

If the editor is unaware of the situation, then you are at risk of committing misconduct.  If the editor knows and approves, you are likely to be behaving appropriately.