It looks like the data signal for both 12 and 5 volt strips can just be 5 volts.
So try, with good possibility of functioning and not much likelihood of damage:
Power the 12 volt strips with 12 volts
Power the 5 volt strips with 5 volts
Feed the signal from the Arduino through a 470 ohm resistor to Din of the first strip
Chain the strips connecting Dout to the next strip Din
Tell us that worked.
[EDIT] looks like I under thought this a bit. Typical. Typing before coffee, never a good idea.
Feed the Arduino signal only to the 5 volt LED strip, not a 12 volt.
When you hook a 12 volt onto a 5 volt, the logic may be ok.
When you hook a 5 volt on to a 12 volt, you may need a simple voltage divider to convert the 12 volt logic signal (if indeed it is) to 5 volts.
AND if you hook a twelve volt onto a 5 volt and the logic isn’t OK, then you would need to convert the 5 volt logic signal to 12 volts or whatever 12 volt strips take. Since these are high speed signals, you’d probably want an active level converter that would pull the signal to both extremes (two transistors), not just a follower.