So...you're both sort of correct, just depending on how you look at the question.
Curing vs Fermentation
First, let's have a quick science primer on food.
All of the food humans consume and utilize for energy can be broadly categorized as differing ratios of protein(remember these are amino acid (AA) chains), carbohydrates(sugars and fiber are types of carbohydrates), water, and fats (lipids). Plants, compared to animal flesh, are composed of different ratios of these classes of nutrients.
The difference between curing and fermentation resides in how these basic nutrient blocks change over time (whether it's in plants or animal flesh), and what chemically active ingredients/microorganisms are added to accomplish this.
What does curing do to these components?
Curing is specifically changing the amount of water that is in the cells of the food in question. Adding salts (NaCl/nitrates, etc), or sugar to water (called a brine), or directly to the surface of the food tends to decrease the amount of water in the cells. This process exchanges the cellular water in part for the added salts/sugars/etc (also called decreasing the water activity). Decreasing the amount of water in the food inhibits deleterious microorganisms from setting up camp and spoiling the food.
Coming back to a part of the semantics portion of your question... you CAN cure both vegetables and meats. The ultimate food outcome and success of curing comes down to the amount of curing compounds, water, time, and temperature. Variations in those components impact the texture, flavors, and edibility of your final product.
OK, so what's the deal with fermentation?
In fermentation, particular microorganisms are either added, or wild-harvested to the food in question. The goal here is to pre-populate your food with beneficial bacteria so that the bad bacteria (species that can make you sick) do not have a chance to get to any of your food. Depending on the organism, they utilize the food-substance to manufacture (called metabolism) varying compounds like lactic acid or ethanol. The chemical byproducts of the bacteria contribute to the texture, flavors, water activity (because the bacteria need to use water to live), and pH of the fermented product. Eventually, through aging, water evaporates or is converted to alcohols, and is used by the bacteria until there is so little water or food substrate left, there are very few (if any) bacteria left alive (they also go dormant) on the food.
Fermentation needs microorganisms, curing is changing the water inside the cells(plant or animal) So you can both cure and ferment meat (even eggs!). You can both cure and ferment vegetables, grains, and fruits. Fermentation of proteins does not equal inedible food. Fermentation by microorganisms like salmonella, listeria, and strains of E.coli on food DOES equal spoilage.
Hope this was helpful.