Wikipedia explains that an herb is
any plants used for food, flavoring, medicine, or fragrances for their
savory or aromatic properties.
So when you see this usage, it is not referring to the name of a person, but a special property of that herb.
For example - "Herb Harashi" means, simply, the main herb, namely, the most important herb currently used in your cooking recipe, or, maybe, the most popular herb in your region. In some places in India they refer to "Herb Harashi, Cardamom."
"Herb Hamachshir" means the herb that makes something kosher. The term "kosher" has a meaning besides the halachic one. It is commonly used to mean "OK" or "acceptable". In the case of cooking, the "Herb hamachshir" is the one that makes the food tasty. Usually, it's a nickname for salt.
"Herb Av Bet Din" has an interesting nuance. It means the herb belonging to the father of the law family (In Hebrew, "Bet" can mean "family".) This was obviously a family of lawyers, and everyone in the family was extremely smart. But the father was a tremendous sage who was the smartest of all of them. Hence, he named an herb after himself. The herb is currently known as "sage" (what else?) or in Hebrew, "Herb Av Bet Din".
"Herb Hamagid" refers to the herb that tells a story, or rather, the herb that people speak about. That could be almost any herb, and it's a general reference. It begins with someone tasting the food that you cooked by adding a "secret" herb. For example, I added tarragon to my home-made tomato sauce (You should try it; it's outrageous!) People who tried my tomato sauce raved on it and asked, "What is that herb that makes your sauce so delicious?" I told them. They started adding it to their sauce, and they're friends and family loved it. Before I know it, everyone in my neighborhood is talking about it. That's your "Herb Hamaggid".