We will talk on the semantics of L and Grz later. I just have defined it above.
R is Euclidian means that if xRy and xRz then yRz.
It is clearly euclidian because if there is a straight line from x to y, and straight line from x to z, there is a straight line from y to z. You can forget it because 5 is the only formula we will never meet. So there is some need to be cautious with the vocabulary, to say the least.