Attached is what the wheel looked like when I lost the other two studs.
Maybe it's just the photo, but that looks like a tapered-seat (conical) nut trying to hold on an original equipment Toyota wheel. If so, the wheel has flat seat areas for the flat washers and shoulders of the correct nuts for use with typical Toyota alloys, and it is absolutely incompatible with these nuts; the tapered-set nut will not stay tight, will not apply enough clamping force without damaging the wheel, and will probably not engage by enough threads. The washer of the correct nut is not visible in the photo, and the edges of the stud holes appear to be damaged by having conical nut ends forced into them.
Anyone who charges for their automotive work and who puts tapered-seat nuts in a wheel with flat seats should be charged with something like criminal negligence, since there don't seem to be any professional associations to regulate the practices of mechanics. Sure that seems harsh, but we're talking life and death here and a child can see that the parts don't fit together properly, especially after seeing the right combinations (this is not so apparent after the stud is broken off, of course).
If that really is the right
nut (because it is a wheel with tapered seats instead of the stock wheel, or because the nut really is the right one for a flat seat), and the right nuts were on the vehicle when it came out of the shop, my comments on this obviously would not apply.
If this this the wrong
nut - presumably the last of a set of five which were all wrong - why was it used? Why was it even available to the "technician"? A Toyota dealership used the wrong nuts on my Sienna when they swapped out the winter tires (which were on steel wheels using tapered-seat nuts) for the summer tires (which were on the original equipment wheels that require flat-seat nuts); in this case I know why the wrong nuts were available to them (as well as the correct nuts, which were in a bag in the van), but there was still no excuse for using them.
...Fortunately when I called my home dealership the next morning I had spoken with the service manager who first apologized to me, and said by all means they will pay for the repair. They indeed did without me having to raise a fuss about it.
... I want to thank the manager in person any way.
Thank the guy responsible for running a shop in which the most basic safety-related procedures are not followed? Paying for the repair was not a favour; it was a manager doing damage control, and it is only the start. He still owes you an explanation, another apology for allowing the mistake, and maybe a free service next time you need one. I would want to see him in person... to look him in the eye and ask what kind of shop he was running.
I had to explain the situation more than once to the service manager of the dealership who put the wrong nuts on my Sienna, because the blunder was so stupid he didn't initially believe that his tech could have done that. Although the problem was caught before any real damage was done, I was offered (and accepted) a free next service from them. Too bad it was over the phone (I was 1200 kilometres away from the dealership at the time), because I wanted to see his expression when he heard about it.