When does self-citation become citation padding?
I think this is easier to think about if we drop the "self" qualification. 


  When does citation become citation padding?


After all, people will also gratuitously cite colleagues ("I'll scratch your back so you'll hopefully also scratch mine") or likely reviewers. And I'd say that the criteria for self-citations should not be any different than for other-citation. Why should they?

That said, I stumbled over this part of the question:


  some amount of self-citation is appropriate (and usually necessary) to provide an indication of what one has done in a field


As José comments, this is incorrect. A citation is not there to show what someone (the author himself or someone else) has done in a field. It's there to give context to the new work, to show what has already been done and how the new work relates to prior art. The causal arrow goes from the work to the author, not the other way around.

(Of course, this is contingent on this question being about an article. In a grant application, you need to self-cite to establish that you know what you are talking about. And if you are writing an editorial to celebrate Professor X's fifty years in a field, you will of course cite his work to show what he has done over his career.)

So the question should be: does this citation provide relevant context to the new work? Who the cited authors are should ideally be immaterial. In your example, it would probably make sense to cite the original derivation of the results. And possibly other applications of such results that were similar to the new applications reported on in the present paper, similar enough to be helpful to the reader.

The problem is of course that "relevant" is subjective, field- and situation-dependent. One field may require a lot more context to make sense of a new piece of work than another one. A review article written in a general interest journal will need more background than a submission to a highly specialized journal whose readership can reasonably be assumed to already have a lot of background. And even then, one reviewer may prefer you to give more background, the other to focus on the closest related work.

And all of these factors apply equally well to self-citations and to other-citations. You will usually work in one direction across multiple publications, after all, and so your previous publications will often indeed be relevant for later ones... just as previous publications of people you collaborate with (because you collaborate with people with similar research interests) and of likely reviewers (because reviewers of course should have worked in this field - otherwise, why have them as reviewers?). So a somewhat higher proportion of self-citations, collaborator-citations and likely-reviewer-citations than of other-citations are to be expected, even without any conscious padding going on.

Reviewers and editors will need to make choices and draw lines to curb excesses, in line with the field's and the journal's conventions and expectations. In case of doubt, you can always ask the author to clarify the relationship of the reference to Foo &amp; Bar (2015) to the present work, or to "consider removing the reference in the interest of brevity".