I think the answer is yes, but you should aim higher: the line you're drawing in the sand is uncomfortably close to you and that's causing more work for you than is necessary.
Namely, trying to put the essential content of your papers on your own website for the purpose of freely disseminating your work seems wrongheaded: instead, you should be putting your own papers on your webpage and/or freely available preprint servers like the arxiv. In order to do so fully legally (which is not necessarily the same as ethically: it is possible to feel that the law is unethical, in which case the ethical thing to do could be to break the law, but be willing to live with the consequences of that) you need to arrange this as a condition of the publication. This is absolutely feasible with all of the major scientific publishing companies. How common the deal is seems to depend on the individual journal and on the standards of the field. In my field (mathematics), I do not know of a reputable journal which would not allow its authors to post on the arxiv. (Once I dealt with a journal which initially asked me to remove the copy of the paper from my homepage. After some back and forth, the editor in chief told me that it was absolutely okay for me to do that and he was surprised that the editorial assistant had said otherwise. In retrospect, this was a clear warning that I was dealing with a shady journal.)
I gather that in some scientific fields, there are "popular" journals which do not allow authors to put papers on the arxiv (or, alas, there may not be an arxiv or clear equivalent) or -- gasp! -- on their own homepage. I find this strange, because in my understanding in most scientific fields the majority of journals are owned by Elsevier, Springer, Science Direct... -- i.e., by enormous multinational, multibillion dollar publishing magnates which are not renowned for their generosity or enthusiasm for open access. But when you publish a paper in (e.g.) mathematics in one of these journals, the copyright notice that you sign allows you to post the paper on the arxiv and your own webpage. I find it hard to believe that the copyright notices for different journals by the same publisher would be so different on this point. So the battle has already been won in these hardest places. If there are trade journals specific to your field which are less generous to authors than the evil empires I've mentioned above: well, if you care about this sort of thing, don't publish there.
In terms of putting copies of published papers on the arxiv, in light of the comments above let me share my understanding. What you publish should not include any journal-specific formatting: it should not look like a Journal X publication. As a general practice, most authors upload to the arxiv before submission, and then only upload a new copy if some kind of significant, content-related change was made. In particular, if an error was pointed out, then it would be good to correct that. If the journal did copyediting for you -- which is not the same as directing you to do copyediting after acceptance -- then maybe it is best all around not to incorporate those changes in the freely available version: on the one hand, aside from formatting, this is the only place in which the journal itself is contributing to the paper; on the other hand, going back and manually incorporating the copyediting for an entire paper could take some time and effort.
To respond to what was said in the comments: in my opinion, whether your revisions were motivated by a referee report should not have any bearing on whether you want to change the freely available version. If you are making changes in response to a referee report, then including that information in a "comments" section on the paper would be a classy move. But as a frequent author and referee, I feel strongly that the changes you made in response to the referee report are not proprietary to the journal in any way (or in any other way from the rest of the content of the paper).
Added: The task of checking out policies of publishers and journals with regard to preprints is not something that an individual researcher needs to address from scratch: there are several online repositories of information about this. Here is one.