Are Liminality and Withdrawal the same?
No. Liminality is one of the main stages of MLC.

1. Separation
2. Liminality
3. Rebirth
4. Reintegration

Withdrawal is an action. It happens many times in different places throughout MLC—such as alienator withdrawal which happens in the early days, weeks and even months after the breakup; that sort of withdrawal is the addictive type. So it is not a conscious action to withdraw into one’s Self or away from someone or something. Alienator withdrawal is the physiological and psychological reactions to breaking contact with the addictive source.
The withdrawal you are talking about is considered a stage of MLC according to Jim Conway’s version of the stages. He places it at the end or after Depression—it may run during Depression, it may follow or both.

During this time, they will NOT communicate with ANYONE, not even their spouse, as they are drawn so far within, no one can reach them. They MUST be allowed to continue, with NO interruptions, just like before-they will NOT come out until they are READY to come out. Just like in Depression, they want to left alone, still processing their issues and the damage they have done to their spouse and their lives, and they make several decisions during this time concerning their lives, job, and marriage. But those WON’T be known UNTIL they break Withdrawal and talk to their spouse the first time.
-HeartsBlessing

Then how then how do they do their job? How do they live or Deal with the people they live with?
You are taking the statement too literally. MLCers compartmentalize. They will not communicate about what is going on inside their head—the inner journey. Some will withdraw more severely and communication with others outside of the journey will also be limited. That may be an attribute of introverts because it is how I am during Liminal periods. At work I avoided eye contact as much as possible—I deliberately looked at my shoes walking upstairs so I didn’t have to acknowledge anyone; I left at lunch or just worked through lunch and left at the end of the day. Normally I’m bubbly, lots of eye contact and I ate in the communal kitchen.
So an introvert may withdraw during Liminality. So might an extrovert, but it may be more severe for an introvert. But that is just an action. It’s a reason I don’t like the labels Denial, Anger and Depression for stages of MLC because they are not isolated to specific times during the crisis. The same is true of withdrawal. The Go side of a Touch-n-Go may be seen as withdrawal. The Distancing in the Pursuit and Distance dance may be seen as withdrawal. Cycling is about coming and going, as well as shifting moods. I even refer to Liminality as an inward retreat—another term for withdrawal.

In MLC withdrawal is everywhere; personally I wish people would forget it as a stage because it causes confusion. When someone’s MLCer withdraws they think he’s at the end of or even past Liminality when it still Replay, maybe even early Replay. Most others come to the person and explain it as cycling, distancing or a Touch-n-Go and the issue is dropped. But as the MLCer gets deeper, when they are approaching or in Liminality, the Standing spouse goes on alert again.

If they are in Withdrawal/Liminality how then do you then lead them along if you are supposed to leave them alone? How do you know when is the right time?
You don’t. It is not your job to lead them along. It is your job to let them go through their crisis as they choose to go through it—and that includes their choosing to avoid. You can facilitate conditions to help them along, but they may resist and that is their choice.

What are those conditions?
• The Unconditionals: Grace Agapé and Forgiveness.
  Nonjudgmental
• Release: Detach, Let Go, Surrender, Acceptance
• Boundaries
• Be the Example: Self-Focus
  Move. Standing isn’t Still. Your MLCer will see you moving ahead and this will eventually be a motivation for their own forward movement.

How do you deal with the anger when it’s directed at your children not you?
If your children are adults, they need to learn to deal with this. Listen and validate your children, but so not fight their battles for them.
If they are almost adults, they need to learn to deal with this and you may give them guidance.
If they are children, protect them like a Mamma Bear. If being with their MLC parents is unhealthy restrict visitations—go to court if you must. Be available to pick them up early from visitations and give them an open line of communication for contacting you at all times.
Give age-appropriate information to help them love through the anger they are feeling and the anger they are receiving. Adults and older teenagers can understand MLC basics; young children will need other explanations. All need reassurances of love.

Why do MLCers run back into replay behaviour?
There was an excerpt included with this question. But some of the answers are in the sentences just before and after, so below I have included the portion excerpted with the sentences around it. I have bolded the original excerpt.

[Liminal Depression] is the place he has been avoiding throughout this crisis and thus there is a likelihood that he will hang on the chasm’s ledge and try not to fall. The depression gets deeper with the Liminality phase when the old Self dies to create a new Self—Ego-Death. The MLCer has no Self; he is suspended in nothingness. It is no wonder they cycle by running back to Replay behavior. The start and end of stages are transitional places where the MLCer exhibits characteristics familiar to both stages. The cycling gradually diminishes at a rate that is dependent upon the degree of wounding buried in his Shadow. Those with greater wounding have more to avoid.
Once a person is in Liminal Depression and has fallen to Rock Bottom their returns to Replay behavior are without his metaphorical skin; they are like a molting crab and highly vulnerable; Liminal Depression is a more comfortable place in this condition.
-Article: Liminality

They rarely run back to Replay once they are fully in Liminality. But before they fall to Rock Bottom they are still trying not to fall—they hang on the chasm’s ledge. It’s not as much a running back as it is that they have not yet progressed forward enough into Liminality that they are at a Point of No Return; they are still Cycling.
It is easier to stay where you are than exert the energy to change. So once a person has fallen to Rock Bottom it is easier to stay there than it is to muster the energy to somehow climb up the chasm and run back to Replay. And when in Replay, it’s easier to stay in Replay. But at the transition where a person is hanging onto the cliff nothing is easy. Think about that in a literal way for a moment. You are hanging over a deep chasm, digging in your fingers in rough ground to maintain your grasp. What is your goal? Do you want to hang in there, fall or scramble back up to there to the ground above? You know what is up there—you were just there. It’s a long way down to the bottom, you don’t know if anyone can survive that fall. And as for holding on, who can hang indefinitely—how will you eat, sleep and relax your tired muscles? Only the suicidal would choose something other than scrambling back to the ground above.

Comments, Questions, Maybe some Answers…?
Ask here in the comments section or visit the thread at the forum and participate in the discussion. I may make this a series if there are more questions.