Thanks DC! That's an interesting idea... but yeah I was really going for the opposite here. In fact I was reminiscing of a great whole grilled lamb shoulder someone cooked for me recently, only trying to replicate that with a smaller cut. But yes, definitely, I like it red inside, not just pink.
I only "marinated" the steak for about 1 hr and I only squeezed a few drops of lemon juice so I don't think it had time to go through to the inside of the meat? I suppose it's really the broiling was much more intense than I would have imagined. I have used electrical broilers to finish gratins or add color to something, but never to cook a piece of meat, and this broiler I have is a powerful beast. I'll have to adjust and learn.
But thanks for the comments! Always appreciated.
I think if I do this again, I'll follow your advice - but still maybe with the broiler (which allows me to lather the steak with mustard, which would make a mess in a frying pan). If it's a thick piece like the one I had (about 2 inches?) 3 mn on each side, let it rest 6-7mn and enjoy. That should do the trick.