English is kind of slippery with such concepts. The difference is whether you want "doing laundry" to be an example of a type, or a sub-type.
Number 1 is indicating that "doing laundry" is an example of a type.
Number 2 is indicating that "doing laundry" is a sub-type.
English is not especially careful about keeping these two things distinct. It is, as in this case, not rare for a sub-type and an example to be expressed in ambiguous form.
Consider a less ambiguous example.
This ostrich is a bird.
An ostrich is a type of bird.
These have been made less ambiguous by adding "this" or "an" at the front. The word "this" is specific while the word "an" is not.
You could do something similar. You could write your examples like so.
- Doing this load of laundry is an example of a household chore.
- Doing a load of laundry is a type of household chore.