It's rp_pi_pico
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Yes! The intended device distinction is made only so that most users would find their firmware project in the correct category under the browse tab.
The flashing tool sends out a reboot command after writing the firmware image, so this is indeed something that should happen automatically. As far as I know both RP2040, and RP2350 won't boot unless a viable firmware image is found, and will re-enter the bootloader instead. Please check that you're flashing the right image. If you're using boot signing, check that you've properly signed your firmware.