How do we define if a Turing machine goes to the right or left? In my college course notes, we are given two examples.


  Example 9.1.1:
  $$ M(K, \Sigma, \delta, s) $$
  where
  
  
  $ K = \{q_0, q_1\} $
  $\Sigma = \{a, \# \}$
  $s = q_0$
  $ \delta = \begin{array}{|c|c|c|c|}
\hline
q &amp; \sigma &amp; \delta(q, \sigma) \\ \hline
q_0 &amp; a &amp; (q_1, \#) \\ 
q_0 &amp; \# &amp; (h, \#) \\ 
q_1 &amp; a &amp; (q_0, a) \\ 
q_1 &amp; \# &amp; (q_0, R) \\ \hline
\end{array}$
  
  
  Note that state $(q_1, a)$ cannot happen if the start state is $q_0$. This is included only for completeness (to make $\delta$ a total function).
  
  This machine will scan right, changing any $a$ that sees to a $\#$. When it first hits a $\#$, it will halt. 





  Example 9.1.2:
  $$ M(K, \Sigma, \delta, s) $$
  where
  
  
  $ K = \{q_0\} $
  $\Sigma = \{a, \# \}$
  $s = q_0$
  $ \delta = \begin{array}{|c|c|c|c|}
\hline
q &amp; \sigma &amp; \delta(q, \sigma) \\ \hline
q_0 &amp; a &amp; (q_0, L) \\ 
q_0 &amp; \# &amp; (h, \#) \\ \hline
\end{array}$
  
  
  This machine will scan left until it encounters $\#$, then halt. 


One TM goes to the right while the other goes to the left. Obviously this is determined by the transition rules, but how do we explicitly know/define where to start (either on the left of the string or right?).

I am asking because I have a Turing machine that works for a problem I have, but I must start from the right of the string and scan left... How do I define this?
It's not specified in the description you gave, but absent directions to the contrary, a TM is generally understood to start at the leftmost non-blank cell.