I wrote a paper in which I mentioned all citations for definitions that are not mine, and did not cite my previous paper's definition or my supervisor's previous papers as both of us are authors.
My supervisor got really angry on me for this. Did I deserve it?
It appears that you mistakenly assume that citations are only about avoiding plagiarism. However, the main purpose of citations is to document the provenance of ideas, evidence, etc. If done properly, this automatically wards you against any accusation of plagiarism, but that is only a side effect. In most papers, you can leave out a considerable portion of citations without introducing any plagiarism (it would still not be a good idea though).
If you treat citations as documenting provenance, it is clear that it shouldn’t matter who authored the work you cite: Ideally, you cite your own work if and only if you would also cite it if authored by somebody else.
Typically, authors tend to overdo self-citations, but you can also underdo them. Usually this is just bad citing, but at worst it can also be regarded as self-plagiarism, i.e., selling your own old work as new work.
Yes, you do need to cite your own past work. Not doing so is considered to be self-plagiarism. While ethically self-plagiarism might not be as bad as plagiarizing someone’s else work, it is still not okay.
Self-plagiarism misleads the reader as it presents previously published work as new and original. Academic readers expect that all uncited material is either common knowledge or a novel idea. By not citing your previous work, you are undermining the integrity of academia.
Next, self-plagiarism is bad because publishers (including academic ones) sometimes have a copyright on your previously published work. While you are still an author you are no longer allowed to use the previous work willy-nilly. However, this also depends on what copyright agreement you have and nature of the self-plagiarism (e.g. paraphrasing old work would not violate it).
Lastly there is no reason not to cite yourself. Self-citations still improve your citation count. You should not do it unnecessarily, because overly self-citing is also frown upon, e.g., even when the previous work is not much relevant. However, when there is a legitimate reason to self-cite, there is no reason not to do it.
Your choice to cite a paper should not depend on who wrote the paper. It does not matter if you wrote it, your supervisor wrote it, or a nazi wrote it. Cite all sources you use. Where feasible, cite other relevant sources.
Do not cite your own work if it's not relevant.
I cannot think of any situation where it would be appropriate to be angry about citations.
