This is a picture of my ceiling. I would like to hang a swing that will hold an adult.
I'm not constricted for space, so I could attach directly to one of the 2x2's
The answer with I-joists is always "ask the manufacturer." Your image's "Trus Joist TJI 210" is sufficient to figure out the manufacturer, Weyerhaeuser. Since there appears to be no blocking nor braces between your joists, the joists are vulnerable to side loading caused by the swing's dynamics. Your installation requires some kind of bracing between two joists at the minimum. It seems reasonable to impose on this bracing to also serve as the anchor point for the swing. That gives you much more freedom in locating the swing's exact position and solves the bracing problem all in one goo (sic). I wouldn't anchor to the I-joist itself. The fastener strength would be sensitive to whether or not you hit the notch at center, and the weight's local stresses on the glue joint around your anchor point would make me uneasy.
You should implement Detail H2 from their documentation:
Specifically, you want to implement the connection on the right twice, once for each end of a solid wood block between the two joists. If the joist is 9-1/2" tall, then you want 2x10 material for the block. 11-7/8" tall? Then you want 2x12 material. It's important that the top edge of this block gets installed as close as possible to the underside of the floor sheathing, but something like a 1/4" gap is still good enough. It's important that this block fits tightly in between the two joists with minimal gap at its ends. Perfectly tight is ideal (not so tight that it's bulling the joist out of the way), but maybe a 1/16" gap is good enough. A gap here has a good chance of squeaking. Many people would install 3 blocks in line with each other, installing the swing to the center block. Given quantities of material that you'll probably be stuck buying, the additional 2 blocks would probably just cost extra labor.
LUS26 joist hangers are the most popular on the market, and those will work fine for joist hangers. You don't need tall ones or anything.
For the backer block sizing and installation I'll put the screenshot from their docs:
Your joists fall under the 210 column, where I assume you have 9-1/2" or 11-7/8" joists. In that case, your backer blocks (you need a total of 4 of them) should be 3/4" OSB or plywood. Footnote 1 recommends 12" long backer blocks, where you cut them just tall enough to fit between the top and bottom 2x2s. A bit of a gap between the top 2x2 and the backer blocks is encouraged to prevent people from damaging the joists by beating the backer blocks into position. You want it bearing on top of the lower 2x2, though. 15-0.131"x3" nails per joist to secure each pair of backer blocks.
The "clinch nails when possible" from footnote 2 means to bend over any nail points that poke through the other side. The 0.131"x3" nails are also perfect for toe nailing through the joist hangers. They would stab through the joist by an extra inch or so, but you might just use the same nails everywhere. Clinched nails are stronger, so you'd only be improving strength by using the too-long nails.
A quick hack would be to screw a full 4'x8' panel of 3/4" OSB to the ceiling running perpendicular to the joists. That's for 16" spaced joists like yours appear to be. Instinctively I would want something like 7/8" OSB for joists spaced at 24", but I can't source that material in my locality. I can source 1-1/8" plywood sheathing, but that's getting heavy and expensive for a "quick hack." You can hang the swing from the 3/4" OSB panel. The side loading has to rack 5 or 6 joists in parallel instead of just the one. As the joists rack sideways there's some extra stiffening from the OSB panel as the racking tries to deform the panel into a wavy shape.
Fasten the panel to the ceiling with 2" screws every 8". For hanging the swing, just bolt a bracket to the panel before hanging it. Something like this D-ring anchor would work nicely.
I would install the bracket at about the 1/4 point between 2 joists (1/4 times the 16" spacing to get a 4" offset from a joist) to instinctively maximize stiffness. Nothing rational about the 1/4 spacing, just a guess.
For sure you can ask people who know about structural wood to advise whether any one attachment point is strong enough. Answers on here already cover that part.
For a suspension hard point though, you should not be looking at a single point of attachment. The relevant people to ask are climbers or anyone else working at height with harnesses. Best practice is for there to be 3 anchor points, going to the carabiner or other fitting from which your swing will be attached. You can build all this in with thin metal cables spreading the load.
It is acceptable for one anchor point to be the main one; but the other two must be able to take the shock load if the main anchor point fails and they have to catch it.
Your method of attaching to beams also needs to be considered. Wood screws are not generally advisable. Best practice is to put bolts through, with suitable plates on each side (or at the very least wide washers) spreading the load over the surfaces of the beam.