Why the internal pullup? Is it needed for this module?
There's no need to worry about pull-ups, either at the Arduino or at the relay board.
Floating conditions (when the Arduino's pins are un-configured) will have no effect and will not turn on the relay because this is an optically controlled (and isolated) device.
Remember that logic 0 (LOW) at the Arduino turns on (energizes) the relay, logic 1 (HIGH) turns off the relay.
However, resistor (R4) will be needed as in your previous diagram to limit the current through the IR led of the opto-isolator. This isn't connected as a pull-up, it's connected in series as a current limiting resistor.
EDIT: R4 not needed, already part of the relay board.