Our small survey of knowledgeable people had the following results. Most (including talmidei chachamim) have never heard of the practice you mentioned. A few follow it. Others are “wishy-washy” about what they heard or do. We hope that some more knowledge will help clarify matters and put them in perspective. There are two possible reasons to refrain from serious work before Havdala. One is a weak (or mistaken) halachic concern. The other is a minhag of classical origin, but one that was apparently not widely accepted.
The mishna[1] says that, as the end of Shabbat draws near, one may walk to his field for the purpose of guarding it and may return after Shabbat carrying fruit. The gemara[2] is troubled how he could do such work (picking fruit) before Havdala, and it arrives at the conclusion that work is permitted after saying HaMavdil. Rashi[3] says that reciting HaMavdil, which is a partial fulfillment of the requirement of Havdala, is sufficient mention of the departing Shabbat to allow work to be done before full Havdala over wine. The same, he says, is true after saying Ata Chonantanu in Ma’ariv. Although both the Rosh[4] and Rambam[5] have a somewhat more stringent approach, the Shulchan Aruch[6] and the accepted practice follow Rashi’s opinion.
Is there a distinction between different types of work? Rabbeinu Yerucham, [7] when discussing these halachot, says that the need for HaMavdil applies only to “work like weaving and writing, not to lighting a candle or carrying.” The Rama[8] cites Rabbeinu Yerucham as a minority opinion, yet the Taz[9] defends him. The Taz hints[10] that the prohibition on work before Havdala is not a continuation of Shabbat’s prohibitions but an independent problem of starting the week’s work before “saluting” Shabbat as it leaves. This idea helps to explain Rabbeinu Yerucham’s distinction. According to him, simple work, even that which is forbidden on Shabbat, is not a problem even before saying HaMavdil. We do not accept this opinion and forbid any type of Shabbat violation before saying HaMavdil.[11] On the other hand, the Sha’ar HaTziyun[12] mentions one opinion that uses Rabbeinu Yerucham’s distinction to derive a stringency: exerting work is forbidden even after HaMavdil. The practice you cited may be based on this opinion, making it a very stringent halachic opinion, or it could be a mistaken application of Rabbeinu Yerucham.[13]
Now, let us explore minhag. The gemara[14] says that he who does work on Motzaei Shabbat will not see good fortune. It is clear from both context and language that this is a minhag, not a halacha, and it is not cited in the Shulchan Aruch. Tosafot[15] and the Tur[16] limit the minhag to waiting until the end of Ma’ariv or Havdala. If the work the gemara refers to is serious “weekday-like work” (as is logical), this minhag could be the source of your mysterious practice.
Given that refraining from exerting work before Havdala is a mistake, an extreme position, or a sparsely kept minhag, you can decide whether you want to follow the practice, especially in your circumstance. (If yes, state that it is b’li neder.) We do not recommend to one who never followed the practice to feel a need to start now. You can opt for the Kaf HaChayim’s[17] approach as a compromise. He says that HaMavdil was intended to allow a woman “to do temporary (ara’i) work, but she is obligated to hear a proper Havdala.” In other words, she should not get overly involved to the point that she forgets about Havdala, but she can act to make productive use of her time until her husband returns for Havdala. (Regarding your specific question about laundry, one can also distinguish between sorting and scrubbing and simply dumping clothes in and turning on the machine.)