I was thinking the following: suppose that one writes an article or lecture notes where he or she uses some lemma. I have understood that every lemma should be used somewhere in the document. But is it enough if I use some lemma in an example, or should there be a theorem or proposition that uses that particular lemma?
But is it enough if I use some lemma in an example, or should there be a theorem or proposition that uses that particular lemma?
I don't think it really matters. An example would be fine for most readers, while a few really formal people might disapprove. However, even someone who disapproves wouldn't consider this seriously problematic. I.e., they wouldn't think less of you or negatively evaluate your paper because of this. The worst case scenario is that someone complains about it in a referee report, and even then you wouldn't have to follow their advice.
The typical convention that I have seen is for a lemma to be a theorem that is only of interest as an "intermediate" result on the way to another theorem. Well-organized presentations thus often use the lemma/theorem distinction to highlight which results are most significant within the material.
There are, however, exceptions to this typical practice. For example, Urysohn's Lemma is an important result in topology that is nonetheless referred to as a lemma. I suspect that such examples typically appear for historical reasons (i.e., the result was originally presented as a lemma but turned out to be more important than the associated theorem), though I do not know for certain.
As far as I understand, a lemma is a mathematical statement that does not have an aesthetic value of its own, and exists only to serve a technical role in proofs of other more interesting statements. The usage of a lemma need not appear in the same document, or in the form of a theorem. For example, it would make total sense for a paper to say "the main result of this paper is a lemma that we think would be useful for proving some fancy theorems in the future. However we still don't know how to use the lemma, and reserve the fancy theorems for future work"...
The question you should ask yourself is whether this is a technical statement that has no internal beauty but can be very useful for other proofs. If this is the case, you can call it a Lemma. However if it is aesthetic and/or unravels some deep truth, it should be a theorem (if its proof is long or complicated) or a proposition (if its proof is short and elegant).