Actually I suspect I know the problem. You ran FS2_Open then transferred it, right? And then you installed on the desktop to a different folder to the one on the laptop, right?
Cause what would happen if you did that would be that the launcher6.ini file would be looking for FS2_Open in the wrong folder and would give you the message The launcher must be in the same directory as the binary you are trying to use."
This is a pretty easy fix. Open launcher6.ini in notepad and change
#
# DO NOT EDIT THIS FILE
#
[launcher]
exe_filepath = F:\Games\Freespace2\fs2_open_3_6_9_debug.exe;
into
#
# DO NOT EDIT THIS FILE
#
[launcher]
exe_filepath = fs2_open_3_6_9_debug.exe;
The problem should then go away when you run the launcher. And yes I realise the irony of telling you to edit a file which tells you not to edit it in big letters.