Is my deck safe or am I overthinking this?

A second story deck that sways side to side when someone bounces near the edge is almost always missing lateral bracing. Tall posts that run straight down to the ground with nothing angled between the post and the beam have nothing stopping them from racking. That little sway you and your brother in law felt is the posts flexing sideways, and it gets worse over time as the fasteners work loose.

Some movement is normal, but not the kind you're describing. A properly built deck that height will have diagonal braces running from the posts up to the beam or joists, usally on both the outer posts and sometimes at the house end too. If you can't see any angled members, that's your answer right there. It doesn't mean the deck is about to fall down, but it means it wasn't built to code for that height and it'll only get looser.

The other thing to check while you're at it is how the deck connects to the house. Look at where the joists meet the wall. There should be a board bolted into the house framing with big bolts or lag screws in a staggered pattern, and metal flashing over it so water can't get behind it. If the deck just looks like it's sitting against the siding with a few screws, that's a bigger deal than the sway. Deck collapses almost always happen at that house connection, not at the posts.

Don't panic and fence the kids out of it, but I'd hold off on big gatherings out there until someone looks at it. A regular carpenter or a deck builder can add the bracing, that's not exotic work. If you want peace of mind first, a structural engineer will do an inspection for a few hundred bucks and give you a written list of what's missing. Given the house is new to you, that report is also handy for negotiating with the seller if the previous owner cut corners. Either way, this is very fixable, usually just some bracing and possibly redoing the ledger connection.

I'd pump the brakes on the 'almost always missing bracing' bit. My old place had a high deck that did the same little shimmy and there was actually bracing on it, the whole problem was the thru bolts at the post bases had worked loose over the years. Tightened them up and swapped a couple that were rusted through and the sway basially vanished. So before anyone starts cutting lumber, go check every bolt and connection up there with a wrench. coud be a ten minute fix instead of a bracing project.

This is helping me figure out my own situation, mine sways a bit too but only when a bunch of people are walking around on it, not when one person bounces. There are angled boards from the posts up to the deck frame on mine, so maybe the bolts thing is what I should check. How do you tell if the posts themselves are just too skinny for a tall deck like that? Mine look like the same size boards as the railing posts and something about that seems off.

Since you just bought the place, worth calling your town and asking if there's a permit on file for that deck. Ours wasn't permitted and turns out the previous owner built it himself, which explained a lot once we started poking around. If it was never inspected, there's no telling what's underneath. Also, mine turned out to be freestanding, not attached to the house at all, so don't assume it's bolted to the framing just because it's sitting close to the wall.

Update: went under there with a flashlight after reading these. There actually are some angled boards from the posts up to the frame, so maybe not totally missing bracing, but the bolts at the bottom of one post look rusty and I can wiggle the nut with my hand. Also called the town like someone said and there's no permit on file for the deck at all. Guess I know what this weekend's project is, then a pro if it looks worse up close.