Since vegan diet avoids animal products, does it include natural honey?
Does it depends whether some bees are killed while beekeepers gather their honey?
Honey isn't considered vegan as it's an animal product made by bees for their own use.
Bees are usually fed a substitute as a replacement which isn't as good for their health, but there are more issues than that; as you've mentioned sometimes bees are killed during removal of honey (sometimes killed deliberately), they are often selectively bred which has health implications and presents serious ethical issues from a vegan perspective.
Some beekeepers do care greatly about their bees for sure (and beekeeping could be its own discussion) but this is about human consumption of bee products.
The UK Vegan Society currently labels honey as non-vegan.
Importantly, harvesting honey does not correlate with The Vegan Society's definition of veganism, which seeks to exclude not just cruelty, but exploitation.
However, it's interesting to note that was not always the case historically!
In 1944, honey was determined by a majority of the Society to be non-vegan (although some founding members continued to consume it). Then in 1948, honey was permitted as vegan. Then it was banned again in 1962, then permitted again from 1972 to 1988.
Aside from the historical perspective, I'd like to add a couple more points of consideration about why self-identifying vegans might choose to avoid honey and other bee products.
Honeybees (apis mellifera) are non-native species in many areas of the world where they are currently kept by beekeepers. They originated in Europe, and are not necessarily well suited to flowers from outside that region. To the extent that honeybees compete with native bee species for food, honeybees may be considered an invasive species.
The role of honeybees for pollination is important to the functioning of large industrial monocultures which are generally regarded as harmful among people who are concerned about biodiversity. A decision not to support beekeepers who use their bees in monocultures may be interpreted as a vote against large agribusiness.