The only thing I disagree with, is about wallets and IDs. Wallets for carrying coinage and valuables existed since the 13th century and were in usage by those well above the "poor" class. Flat Wallets existed carried paper currency or bills of trade, some were just large enough to carry some coinage too. Shoulder wallets or Market wallets, were shoulder bags used for various purposes and as general usage bags.
Modern names for some older wallets we would call: Men's shoulder bags, sling bags, belt bags, reusable shopping bag and fanny packs. Yes, Fanny packs are not new, just a reinvention of a coin purse/belt wallet with new materials.
People did carry Identification such as Calling cards in wallets and on themselves. The "ID" may be a letter of recommendation from someone of status known to another known personage of status to identify the bearer as who they are to another. Or a seal or a badge, for identification purposes. Besides calling cards.
Now as to the question:
I would be well armed and alert for an ambush in such a case coming upon a fat wallet. If it were spotted by the coachman or a footman and I would not necessarily agree to stop and have a look at it. That is what footmen are for. The Coachman would not grab the wallet, a footman would. We all would be all armed "appropriately", the Coachman with a Blunderbuss plus a "walking stick" and Horse whip; myself and the valet will be equipped with my fowling piece and/or brace of pistols; plus pistols and cudgels/long knives for the running footmen.
As to the vagrant. A footman would be designated to give them a few farthings and to have them questioned on the where abouts of the wallets owner. One must be christen when one should be.
Now a calling card and such like inside the wallet, is actually a thing for that time period. That would indeed be a reason not to disburse the wealth in the wallet, but to forward it on to the appropriate personage named in the wallet via the servants later.
If I am inclined to treat with them. No, I would buy them food with it.
Back in the time period set by the OP within London. The way you navigated was with landmarks. Nobles barely got around to it, at the middle of that century, giving their houses numbers. There was no formal system, there was nothing like what we have today.
But those house numbers and everything still heavily relied on landmarks, and relied on people who had money intentionally setting that up, places outside of large cities still didn't have that. Unless we are dating the interaction after 1750, in an upscale, richer part of town or in the areas of newer development buildings, where the postal service started establishing house numbers and their entire system of organization, people would still be more likely to use landmarks.
I agree that IDs at the time would be the different letters of recommendation and a few calling cards, in the setting provided you would have to expect the person to be within your own social standing at that point which means one of their workers is going to end up punished or worse.
There is no chance the commoners would ever have wallets implicated in the original post; since almost everything listed as a requirement is a class privilege. You also still wouldn't hand it off back to them, thats what the steward is for when you return back.
The notion of charity for the noble class would be at structured events by the church, it is beneath your station to give alms on the street, hence handling it through one of the staff and only then when there are others watching.
Paper currency, again in specifically London, was only introduced in around 1690, these were bank issued promissory notes. Things no random person would have on them. Not to mention there was no fixed denominations because they were handwritten. Key point that they wouldn't be for normal transactions.
Sure after middle of the century it might be more common because they shifted into printing bills and currency, but the question was 'stacks of cash', the bank restriction act for converting gold was 1797 when they introduced £1 and £2 notes at the time. Which is about the first chance someone lower class would probably have a reason to see note denominations.
This isn't about existence of bags, as those always have been around, you said yourself however, they are coin bags, not rolled up bank bill bags. The rabble of normal people would have no reason to carry that much money around with them, they wouldn't even really be able to save up that much money half the time.
I will completely agree however about the footmen, but I was originally trying to lend the story more credibility with a carriage which was already adding in an extra person in that context, so the assumption was something more covert, but yes, a proper group protecting the carriage and the driver not leaving his post would be safer. Armed retainers are pretty much a requirement for the social standing given the dark London streets was the original question. I wouldn't have defaulted to sidearms too but yes, flintlock pistols would probably be a sidearm for some of the retainers too.
As for the kid, the option I mentioned would be the most socially acceptable thing, notifying the church and leaving the matter, but yes a more direct token act of charity could work, a few farthings might be slightly too much for a random urchin but I suppose it's your money.
I would expect for the time period however that the footmen are more likely to shoo the urchin away possibly throwing them one farthing rather than interrogate them about the wallet, it would make no sense for the kid to not just take the wallet at that point even if they can't use the notes and whatever else there would be some coins I would assume. Either way there is no reason or expectation for the child to have any knowledge on the wallet if it isn't already stolen and looted. It would even be deemed dangerous for the footman to approach a random child, it isn't really their job to question urchins either.
Being a good Christian and following your duty still didn't involve street handouts and would only be done if in front of other peers, otherwise it would be donations in the name of your house, handled by your staff. Performative charity happens in public and charitable events. In this setting it would be the minimal standard required. Overall the idea of charity is heavily impersonal, handled through the church for the most part beyond small token charity.
My whole point of writing up everything I did before was to highlight that the entire setting felt misplaced, as I missed one or two details in my first message, so thank you for helping expand on that.
People misunderstand a lot of history and it shows when novels follow after other media misrepresent the era, I might be overcorrecting some details but there is also a bias that's worth challenging for when stories depict nobles and commoners somehow share learning institutions during this time period or when there's a massive misunderstanding of social class divide. If we are trying to be accurate, then these things need to be known and established.
If you understand history enough to correct me, I assume you also understand the idea of trying to break things down into something you can explain to others, even if it compromises the full accuracy of what is being said.
Thanks again though for the reminders and clarifications, the more accurate the better when it comes to this type of stuff, right?