Ciao and welcome to our forum!
First of all, make sure you rent a car so that you can then be able to choose any place you find, regardless of its location in a town or in the countryside. It is likely already part of your plan but just to make sure you get used to the idea of driving while here, it will open up all the possibilities of staying anywhere and not just close to a town with rail/bus connections. You don't want a car once you go to Venice so you can plan to rent a car while in Tuscany, then return it in Florence and take a fast train to Venice. Are you flying into Florence, Pisa or Rome?
As far as agriturismo or villa, to be honest the type of lodgings are very similar. Most are born from old rural houses or larger houses on a country estate and the name - whether it is called a villa or farmhouse - just depends on its history. Consider they were all born as country estates that had land and were agricultural in nature. If a wealthy family lived on the estate, it is likely it built a main VILLA and many smaller houses around it. If it was less wealthy, maybe most of the houses on the estate are about the same size, with the main house still being a bit bigger but not overly bigger. Many of those other houses used to house the families that lived and worked on the estate, whether they grew olives and made oil or grapes and made wine. Many grew grains as well. They likely had barns as well as haylofts, many of which today are no longer used to farm or to house animals and which have been converted into housing.
The properties/estates today generally are called "agriturismo" if they continue to have an agricultural business on the property - most are olive oil or wine - while those who have sold the land and maybe not work it directly might not be able to call themselves an agriturismo because of that, but the history of the place is the same. Some are called "podere" as opposed to agriturismo but still have a business in growing on the land around it so sometimes the name itself gives you an indication of its history more than what it offers.
In any case, both types of accommodation are composed of houses that have been divided up and converted into smaller housing units, which either become apartments if the original house was large enough or just rooms to rent in a larger home, in B&B style.
There are villas that offer B&B style accommodation and villas that offer apartments, and then there are agriturismos that offer the same, either rooms or apartments. Most will have a shared pool and gardens, as well as their own external garden area reserved for just that apartment/room.
So rather than focus on agriturismo vs villa, consider whether you want to have two rooms and no kitchen or kitchen access or whether you want two single-room apartments or a single two-bedroom apartment with kitchens. That is where your choice will fall on.
This all to help you and others make a sense of what you see as you search for a place. Since you have a budget, I think I would start from there. Exclude the obviously luxury or high end of the offerings and start limiting your search. There are countless choices in the end. I have to say your budget is pretty high in any case (300x10 is 3000€ total, you should check weekly rates are generally closer to the 2000€ mark) so you should still find a lot of choices!! The more elegant and recently renovated but be on the higher end than the more rustic farm-style furnished places. Most places rent by the apartment and not by person but of course larger apartments generally cost more.
Next method to limit your search will be to decide what you want to do during those 10 days, then select an area to search and thus limit your choices. Since it is your first time in Italy, it is likely you want to include the "top" attractions which include Chianti wine region, Florence, Pisa, Siena and Val d'Orcia. For this reason, I would suggest you look at places anywhere between Florence and Siena which is a really big area generally called
Chianti and it is far wider than the wine region itself. Take a look here to start:
https://www.tuscanyaccommodation.com/chianti/ just to start searching and let me know which ones appeal to you. Once we've narrowed down a little more what you want, I can suggest specific ones.... although as far as view goes, you will be hard pressed to find ones without the great view!
Ok, hope I've given you enough food for thought at the moment, let me know about what you see and like, then maybe we can continue from there!