I believe (based on experience and information provided in another thread on here) that the display of economy class type on other airlines booked using United MP miles is wrong on all LH/LX/OS etc. Essentially you should go by the luggage allotment noted on the united ticket (reservation) and that's what ends up happening at check in.

TL: DR You'll get the first checked bag for free on LX/LH/OS if on an award ticket from United.