Joint-map approach: The quickest way to crack hand-arthritis questions is to ask which joints are hit and which are spared.
Reading this case: DIP and PIP joints plus the thumb base are involved and the wrist is spared. That distribution, with Heberden nodes at the DIP and Bouchard nodes at the PIP, is textbook osteoarthritis, since the DIP is the commonest OA site in the whole body.
Excluding the rest: Rheumatoid arthritis does the reverse, it loves the MCP, PIP and wrist but classically spares the DIP, so wrist sparing argues against it. Psoriatic arthritis can involve DIP joints but is given away by nail pitting, skin plaques and dactylitis, none mentioned here. Gout favours the big-toe MTP joint. So osteoarthritis is the best fit.