Strangely, both of my presta stems broke on the same day, like the stem thing broke off when I was filling. On one of them it was one of the repairable kinds and I scavenged a top from another tube. The stem is rattling around inside... The other was unfixable and I got a new tube. Referencing the pic below (found on another thread), does the "this part unscrews" part need to be tightened significantly? On my new tube, it came right out when I tried to remove the cap. I snugged as hard as I could with my fingers, but I don't have the special tool and didn't want to mar it up with pliers.
I've never seen anyone publish a torque spec for a Presta valve core, and measuring it would be difficult anyway.
While generic in terms of form factor, there can be quite a bit of difference in terms of whether the threads go together smoothly versus having some gumminess or other friction that has to be overcome. I believe some have various kinds of pre-applied threadlockers or sealing material. That will in turn have a lot of impact on how well preloaded the threads are for a given torque input.
Personally I don't like the idea of not using a tool. They do sometimes come out if they're under-torqued, and you don't want that to happen. The "right amount" to me is typically the most I can get without the valve spinning, which still isn't all that much torque.
You can install them with a small (4" or 6") adjustable wrench. Another good improvised tool is a ring-type spoke wrench, or a blue Park one.
There are small plastic tools that you can carry in any bicycle bag or pocket. In my experience, it is enough to be finger-tight with this tool, but just finger-tight without any tool at all brings the risk of getting the core unscrewed inadvertently. Any bike shops will have many pieces of these tools and might even give you some for free. It is really just a small piece of plastic.
There are also other, more screwdriver-like, tools dedicated to Presta valve cores, but they are not necessary.
I've had the valve core back out with the pump head occasionally.
The fix is a small dab of locktite or superglue on the thread, and then tighten it to just-past finger tight. This way, you're unlikely to get a pump head tight enough to overpower the stem's thread.
I don't have great finger strength, so a tenth to an eighth of a turn past finger tight is sufficient. I can still get the valve stem out with a spanner if I want to save it or put some tube sealant in.
Do note - the valve stem does have a couple of flat spots on it for a wrench/spanner. Your photo shows them in the Silver part, right at the tip of the arrow. A spanner here will not marr the threads.