The more recent cases are different. First, none of these people is young James or Lily Potter, and then there's the precedent set by the betrayal of James and Lily by one of their closest friends (kin, in their eyes). Anyone in the Order with need to use the Fidelius Charm would think twice before picking someone else as SK after that (and might even be the reason why recently people began being their own SK instead of asking a third party)

Having Lily be the SK would have been safer, but also a lot more constraining, for people who hated being constrained and put aside.

It's just more practical to have a third party be the SK, as it leaves those in the house the possibility to get out if they must,(to go to Peter, for instance, or to bring help). Lily as SK could not have come out at all, and left James forced to come and go a lot more, a for instance it would have forced the couple to send James out to fetch anyone they wanted at the house, and to have him bring to the house anyone they wished Lily to reveal the secret to. So either James went out a lot, or they trusted someone like Peter to bring people to Lily anyway. Comings and goings were very risky because of potential spies in the vicinity, and because with someone who know, you can come in (as was done to bring Harry into Grimmauld Place, IRRC).

The way they chose to do it also ran a fair chance of misleading Voldemort, who would have thought (if he didn't have Peter, anyway) his only chance was to lure Lily, the obvious SK, out of the house and to capture her. From the Order's perspective, that cornered Voldemort into risking to expose his mole, out of despair, to lure Lily out. If their suspicions about someone being the mole became great enough, Lily could even come out and make herself the bait to expose the spy or trap the DH, which she could never do if she were really the SK (much too risky for Harry).

The Potters rather did something no one expected and that a lot might have thought way too reckless: they trusted someone else to be their SK, and that someone else wasn't even Dumbledore (another obvious choice for Voldemort to consider, and another near dead end for him).

Peter as SK meant both Lily or James (probably never both at the same time) could have the option of leaving the house if necessary, at high risk to themselves for sure, but without endangering Harry badly every time they did. That at least preserved for them the illusion the Order could call them if they were really needed, that they had not been completely sidetracked, and Peter was someone James and Lily trusted to give them the whole picture of what was going out outside. Peter could also be trusted to comply without a fuss when asked to reveal to some people the secret, like revealing it to Petunia, or that woman from outside the inner circle who was visiting Lily.

Meanwhile, as Voldemort put his efforts on this dead end of luring Lily out, the real and unsuspected SK was free to come and go, and be the eyes-and-ears of Lily/James outside. Peter was more low-key than Sirius and Lupin, yet as trustworthy, and he had a way to get to the house totally unseen, and even to go in hiding for years if necessary (all as a rat, of course).

Dumbledore might have been safer, but I get the feeling from their perspective James and Lily did not trust Dumbledore to keep them properly "in the loop" with the Order's affairs if he was their SK, or to call them up if there was great need for them, as their safety would have been paramount to him etc. Dumbledore could have made them "too safe" to James's taste, that is too constrained, IMO - which is why they picked "one of the gang" instead of him.

Picking someone not from the house was risky, but we're dealing with two daredevils who must have disliked very much the fact they had to leave the war and go in hiding.