I have no idea what you meant when you said, "You are confusing wanting to save anyone than thinking it is your job to save everyone." But just in case, I will reiterate what I said so you understand what I'm arguing for.
What would make the difference if they were older and "had proper training"? Sure it would enable them to diffuse a dangerous situation more efficiently, but would that make them less, so to speak, "arrogant"? Training would indeed give them more bragging rights, I guess you could say, but whether or not they have the tools to save a person, wouldn't it be arrogant to think you can save everyone in either situation? (Disregarding the fact that I believe they know they can't save everyone)
You have to admit, trained or not trained, Shinichi was smart to utilize the whole "dropped your bullets" trick, which was a clever way to catch the killer off guard and kick the gun out of his hands (which looked quite professional to me). I don't think you need any formal training if you are able to predict the murderer's every move like a game of chess. Shinichi's greatest weapon is his brain, which is sometimes all you need in order to deter the killer's actions (with the use of psychology). And we could throw the "what if" card in any scenario, but considering how well his tactic turned out, I don't think it's necessary.
You didn't receive the memo? Ran is a bad girl now and wants to troll every situation that comes her way. She's officially a newborn rebel. (That was my failed attempt to redeem myself for forgetting to mention that in my argument) I just watched that episode yesterday, let alone read the manga chapter, so I can't believe I completely neglected it as an example... This is the only case that I said: "What the hell are you doing Ran?!" I'll admit that she was stupid here, but if we take into account all of the other scenarios, including the ones that I previously mentioned, Shinichi and Ran are not selfish for trying to save as much people as they can WHEN THEY ARE GIVEN THE OPPORTUNITY TO. If they were handed the chance to prevent one's death for the sake of justice, then why not? It would be a whole different story if they went about seeking and searching for opportunities to stop criminals and save lives like a comic book superhero. But no, in every case where they did save a person, they were granted the opportunity, not on their own will. So IMO, it's not selfish.