But after self-reflection, I'm more likely to report it if it did have money.
If it had money, I'd feel an obligation to protect it and return it to the owner. If it didn't, I'd feel more like it's their problem.
In only one case do you actually lose money. Both cases require the same effort to make contact.
For the civic-honesty-minded person who has to balance that effort cost to themselves against the victim's loss, there's going to naturally be a stronger impulse to help the person who stands to lose more.
Say I've lost both a house key and also enough information to, in the current era of scummy personal information aggregation websites, find out my home address, which you can definitely commonly do based on just name and email address and an assumption of local residency. Then regardless of whether I get the key back I should in paranoia change the locks on my house, because now an unscrupulous person who found it easily has a copy of the key and knows exactly where to use it.
So if it's a house key, then in defense it should have no more value to me but has significant (hopefully temporary) value to them.