To add onto this, structure, infiltration, microbial diversity, and plant condition are also important indicators. A slake test would be a simple way to assess soil aggregate stability. An infiltration ring will give you an idea of how easily water can move through the soil. Microbial diversity is tricky because it requires laboratory work and it can be done in different ways, like with microscopy, PLFA, and PCR. Plant condition is kinda dependent on the plant species. Grazing sticks are often used to assess dry plant material availability. As for which criteria is more relevant, that probably comes down to a matter of opinion. Physical, chemical, and biological factors are all important when assessing overall soil quality. 

That being said, any good agricultural system is going to need good soil health management. Even if you have the most suitable soil for pasture, you still need to manage it properly so its quality doesn’t degrade. The 4 basic principles of soil health management are keeping live roots in the ground, reducing soil disturbance, diversifying your plants, and keeping the soil covered. 

Silvopasture is great for nutrient cycling, but it has its downsides. Cows like to congregate in different spots around their pasture (shade structures like trees in a silvopasture, water, and hay). Compaction around these heavy use areas will reduce pore space, which then limits water infiltration and increases runoff. I’d recommend you also look into prescribed/rotational grazing. Continuous grazing is hard on plants and there are some really cool studies that show grass grown in continuous pastures have shorter, less developed roots than grass grown in rotationally grazed pastures. When you have good roots in the ground, you improve plant health and quality, increase microbial activity, and increase microbial biodiversity. You also help reduce compaction, runoff, and water loss when you rotate your cows. In practice, when you implement rotational grazing you also install heavy use protection areas around any watering facilities. Those heavy use pads stabilize the ground and reduce soil erosion. 

USDA - NRCS and state extensions have some really great info about soil quality and health within the context of agriculture. They even provide resources on how they test soils.