What is a difference between degrees of freedom (DOF) and degrees of motion (DOM)? I know that DOF is the number of independent movements a manipulation arm can make and robot system can have max 6 independent DOF and unlimited number of DOM but I do not distinguish them from each other.
The way you are describing it, DOM is the number of independent dimensions in $\vec q$. DOF is the number of independent dimensions in $\vec x$. In practice, a robotics engineer will use DOF to represent the number of independent actuators of the robot, which you are calling DOM. Better notation would be to call DOM the mobility of the system, as described in the Mobility formula section here.
Using your notation:
Think of a telescoping pole made out of 5 sections that can collapse inside each other. The degree of freedom is the length of the pole. That value can be anything between its fully-collapsed length, up to its fully-extended length. DOF = 1.
If there is a sliding joint between each section of the pole, there will be four independent motions you can use to set the total length. DOM = 4.
In kinematics, the length would typically be denoted by $x$, and the individual joint lengths would be assigned values of $q_1$ through $q_4$.
$\vec x = f(\vec q)$
It appears the industry need to consider some standardization of the terms. In my search I have found differing definitions for the term DOF.
From Yaskawa/Motoman: "Degrees Of Freedom : The number of independent directions or joints of the robot (R15.07), which would allow the robot to move its end effector through the required sequence of motions. For arbitrary positioning, 6 degrees of freedom are needed: 3 for position (left-right, forward-backward, and up- down) and 3 for orientation (yaw, pitch and roll)." Note: R15.07 would appear to reference the ANSI/RIA/ISO materials but at RIA, I only found reference to R15.06.
From Robot Worx: Degrees of Freedom - The amount of values in a system possible of variation. A robotic joint is equal to one degree of freedom.
From ATI automation another RIA member: "Six Degrees of Freedom - A fancy way of saying Fx, Fy, Fz, Tx, Ty and Tz."
I would have to disagree with the Robot Worx, because at a minimum both directions of rotation or travel must be considered for each plane. So Yaskawa's explanation of 3 planar linear directions and three planar rotations makes more sense as well as the definition from ATI. If this is accepted, then what of DOM? This term must inevitably be more complex and account for many things other than direction. I see no terminology so far that accounts for actual degrees of motion of servos/actuators for example servos designed for 180°/360° motions. There should be some way to relate this functionality in the glossary of terms whether it be DMO or yet another term. Some method of using all the available functions of a robotic device should be somehow described.
In my opinion there should also be a standard definition for a description that includes, such other functions as: time, acceleration, speed, grip/release, etc.