No you don't get to pick the particular Turing machine if by "particular Turing machine" you mean a Turing machine including a particular transition function, which is the conventional definition of Turing machine (
http://en.wikipedia.org/wiki/Turing_machine#Formal_definitio...)
In terms of this definition we need two Turing machines, one whose transition function can be proven to be the shortest that describes L1. And another one whose transition function can be proven to be the shortest that describes T0. These two Turing machines or transition functions are compressed representations of L1 and T0 respectively.
What I'm saying is that the one describing L1 can be shorter than the one describing T0 even if L1 is longer than T0.