You are precisely wrong. You are conflating the fact that the German Nazi government promoted eugenics with the idea that the meaning of "eugenics" changed meaning at that time. Regardless of who chooses to do it or why (or even how), deliberately trying to change breeding habits to people to bring about a desired change in genetics is eugenics.
(Incidentally, at that time eugenics was very popular with other governments too. this was the era when the US and UK governments forcibly sterilised thousands of people in insane asylums, and for offences such as being poor. The testing of immigrants for IQ at Ellis Island was part of this move too.)
I'll expand on that "how" : there are two ways you can produce a change in gene frequencies in a population - you can cull some genes from the population (by sterilisation, or by murder - same outcome) OR you can try to persuade particular "desirable" groups to have more children - which is what Galton proposed in the 1870s and onwards, and deranged idiots like the "Quiverful" movement and the Catholic Church are continuing with to this day.
If you were a farmer, or a dog breeder, you too would be practising eugenics on the populations you control, though the term isn't generally used for non-human animals.