Given a robotic arm, if you derive homogeneous transformation matrix for it , it will be equal to one of the above mentioned equations.
Those equations are the basic scenarios for reaching the end point, any robotic arm will satisfy one of the three equations .
The equations simply mean the order of manipulations carried out by the arm.
A translation along x axis followed by a rotation
A rotation followed by a translation .
If you observe the equations you can notice that there can be two consecutive translations but not rotations.
These are the basic scenarios, however for a custom robot the equations can be different
For example,let us consider a robotic arm with a revolute base followed by prismatic joint, and another revolute joint
The scenario of this robot is
If another robot has a prismatic base, revolute joint, and again a prismatic joint, the equation would look like
So, it depends on the robotic arm that you are working on. But these are the basic situations to understand that sometimes translation is followed by rotation or vice versa, or two consecutive translations but no consecutive rotations. If you understand these equations, you can build your own using them.
After reading back and forth in my robotics and modelling book, i forgot to tell you one thing in DH notation matrices.
The Transaltion matrices Trans(z,d), Trans (x,a) are identity matrices with just one vale in the last column.
1 0 0 0
0 1 0 0
0 0 1 d
0 0 0 1
1 0 0 a
0 1 0 0
0 0 1 0
0 0 0 1
So, they dont change the overall result even if they are multiplied before or after the rotation matrices.
I used the second equation to calculate transformation matrix for puma 560 in a java program. It was perfect.
I hope this clears your confusion, sorry for the delay.