I think your definition for the term pesticide is perhaps too narrow.
Per Saunders Comprehensive Veterinary Dictionary, 2nd Ed., pesticide is defined as (p. 865):
a poison used to destroy pests of any sort
And it defines pest (p. 865) as
an organism that injures, irritates, or damages livestock or crops
Therefore, I think your answer for the general word is pesticide.
See also this comprehensive definition in Wikipedia for pesticide.
Pesticides are substances that are meant to control pests or weeds. The term pesticide includes all of the following: herbicide, insecticide, insect growth regulator, nematicide, termiticide, molluscicide, piscicide, avicide, rodenticide, predacide, bactericide, insect repellent, animal repellent, antimicrobial, fungicide, disinfectant (antimicrobial), and sanitizer. The most common of these are herbicides which account for approximately 80% of all pesticide use. Most pesticides are intended to serve as plant protection products (also known as crop protection products), which in general, protect plants from weeds, fungi, or insects.
There is also the word biocide, which is not as much in general use. It is defined by Merriam-Webster as:
a substance (such as an algicide or fungicide) that destroys or inhibits the growth or activity of living organisms
It is a term used by environmental groups, such as in this post by the Green Party of California.