Sounds like there are traffic initiatives at LAX. While the "taxi full length" would literally mean to taxi the full length of the runway, it's not something that you hear at every airport (you'll usually hear "taxi to runway 2-8 via taxiway Echo, Golf, Hotel, Hotel One... etc). Someone familiar with LAX operations may be able to clarify why the "taxi full length" is used.
As for the takeoff, when you hear "<Callsign>, runway 2-8 FULL LENGTH cleared for takeoff" that means the aircraft is LOCATED at the end of the runway and that aircraft is cleared for takeoff. When you hear "<Callsign>, runway 2-8 at Bravo (or other intersection), cleared for takeoff" it means that the aircraft located at that intersection is cleared to takeoff from that intersection.
The difference?
Controllers must use these terms when one aircraft wants to use the entire length of the runway for takeoff and another only wants to use a certain length of the runway (larger aircraft needs more runway to takeoff than a smaller aircraft that were both taxied to the same runway for takeoff). If they have two aircraft holding short of the same runway (one at the end and another at an intersection), they must mention either full length to the aircraft at the end of the runway or the intersection name to the aircraft departing from an intersection.
It's not exactly easy to understand right away and it's easier when you can both see and hear the instruction in use at an airport to better understand, but I hope this helps a little bit!