What is the difference between
MENUPETZ LISHMA THICK and
AVODAS YAD THICK?
In the process of making the tzitzis strings there are a number of stages. The first one is called niputz, which is that the wool is combed to separate the fibers and to remove impurities. After that the wool is spun into threads called tviah (spinning). After that the various threads are twisted together to form the actual tzitzis strings, calle shezira. The final two stages must be made lishma- meaning with the specific intention for the mitzva of tzitzis. Regarding the first stage, the general minhag is that it does not have to be done lishma. Menupatz Lisham means that even the combing was done lishma.
Avodas yad refers to the fact that the tzitzis were spun and twisted together by hand and not with a machine.
The word thick has nothing to do with the process of making the strings, but the thickness of the finished strings. Typically a tallis will have thick strings, and often a talis katan will have thin ones, but some people prefer thick ones even for their tallis katan, (it is stronger and harder for it to rip).