I made the mistake of ordering LR5 from Adobe, who's website showed only download versions of LR5 available. LR5 then appeared a couple days later on retail sites as a hard optical disc copy.
When I went to download a second copy to my laptop, it assumed I wanted the 30-day trial, even after I entered my new, fresh, paid-for serial number.
I am slightly confused by what you say??? AFAIK anytime you launch an installed trial version of LR it asks whether you are trialling or want to buy or have a serial number. It does not ask at the point of download. Also why bother to download a 'second copy' just burn a CD or DVD of installer you downloaded to start with (or copy it to a USB drive) and use that to install it on the laptop
For the record I downloaded, installed started the 30 day trial but in the meantime I ordered the DVD retail version from Amazon UK and once I received that I used the serial number on the package to "license" my downloaded trial.
So can I ask the obvious? Is the version you downloaded the "upgrade" version............if so did your desktop PC have the previous version already installed but the laptop does not??? If that is the case then you will need that older version serial number as well to install it on the laptop
Incidently, my desktop PC does have the older versions and as part of the serial number entry it automatically found the serial number of the older version and pre populated the fields.
I hope that helps but if not please describe step by step what you purchased and what steps you took to install