Some can argue that antibiotics are also a pesticide against pest species of bacteria. You also get rodenticides since mice, rats and other small mammals are also considered pests in some contexts. By the definition of the word a pesticide is a toxic compound used for the control of species considered to be a pest. Pest species tend to be those that compete for resources with plants/animals of agricultural or industrial importance, spread disease, are an invasive species or cause a nuisance. Ideally pesticides should be selective to not harm off-target species but this is seldom the case. Herbicides are compounds that are toxic to plants and are usually used as a pesticide to control plants considered to be weeds. I believe the argument of your mentor is that although compounds in those categories can have application as pesticides, they are not automatically pesticides by being toxic to those classes of organism.

Herbicides are applied to kill plants (e.g. weeds, invasive plants, etc.)

Herbicides are one category of pesticides.

Pesticides are applied to kill pests. Pests maybe plants, they may be fungi (pathogenic), insects (as pest insects = insecticides), mites (as in miticides or acaricides), etc.

Then there is homicide (and its variants), genocide, and so on. The suffix "... "... icide" refers to killing.

On a holistic approach, herbicides are commonly used to eradicate unwanted plants in managed lands ( therefore, weeds and invasive plants are targeted in case of managed agricultural or garden lands). On the other hand, pesticides target harmful (hence unwanted) insects in same land use patterns in order to primarily obtain optimum crop yield.