Simpson makes brackets that can be attached with joist hanger nails that are NOT joist hangers; I have them on the ends of my stair stringers. Most likley the:
L/LS/GA Reinforcing and Skewable Angles
I suppose other companies make similar things, though to some extent SST is such a behemoth I don't know of any others in my local market. Anyway, if the ledger is carrying the bottom load, these will provide reenforcing for the attachment.
They also make flat reenforcing plates that could be attach across the splits to stabilize them, at least temporarily (which can be a few decades in house work, often enough...)
Yes, there are grading rules and tolerances for lumber, timbers, etc.
You are describing "dimensional lumber". Grading rules for that (especially where you live) is established by the American Softwood Lumber Association, PS-20. (Product Standard 20.)
If you look at Table 3, "Dimensional" you'll see for 2 x 10's, the minimum size is 9 1/2" for green and 9 1/4" for dry. (This can vary somewhat based on the species.) ...but the minimum size is 9 1/4".
Where I live, grading is primarily done by Western Wood Product Association. If you look on your joists, you'll see a black ink "Grade Stamp". I suspect you'll find several numbers and letters in a small grouping (about 1"x2" area) with one of the icons of WWPA. Next to that round icon, you'll see the "moisture" grade. If it reads GRN, then it was milled (cut at the mill) when it was green and had more than 19% moisture content. If it reads DRY, then it was milled when it was less than 15% moisture content. (There is a seldom used mark of S-DRY, which stands for Surface Dry and had a moisture content between 15%-19%.) Actually, if the stamp is smeared and "out of focus" to the point you can't read it, then it was milled "green".
I explained all that to let you know that your lumber could have been milled "green", but dried out, which caused the lumber to shrink.
If you read the text, you'll see that there is a tolerance of 1/4" in the width and 1/16" in the thickness...the 2x direction.
Also, the lumber could have been mis-cut and was rejected, so you won't find any grade stamp on your joists. The contractor could have bought the joists at a discount and was never caught by the Building Inspector.
By the way, the grain of wood in your picture indicates a better than a normal piece of lumber. If all the joists are similar, then I'd feel comfortable that it's structurally sound and would equal other lumber that is 1/4" larger.
Best Answer
I can think of a couple ways you might go here. The rim joist doesn't carry much actual load. Most of it is on the common joists. Really, you can remove it and replace it without doing much else. Keep an eye on things as you do so and watch for any twisting movement in the in the joists.
Another approach would be to rip cut the joist halfway down and replace it with two boards. Remove the top half and replace it with a 2x6, then remove the bottom half and replace it with another 2x6 ripped to the correct dimension.