What is the recommended post cap or strapping for corner post under beam and rim joists?
If you are butting a rim joist into the beam, you can frame like the first photo with the beam held back 1-1/2" and the first rim joist extending to the end of the post.

Then, in the corner of the second (inner) rim joist and the beam, attach some Simpson A35's.  1 should do, but if you have enough height you could install 2.  Each A35 can transfer nearly 600lb of gravity load into the beam.  If this truly is a rim joist, then 1,200lb of (2) A35's + the shear on the first rim should be more than enough.

Then use (6+) 3-1/2" quality construction screws (vs cheap deck screws) end screwed through the outer rim into the end of the beam to tie the rim and the beam together.

I would not recommend a Simpson ECC style column cap; these are only needed in extreme (for DIYers) load cases.