I think there is a difference.
I can't find the definition you wrote, "is to deal with a successful situation or problem", in the source you linked. The second meaning of "manage" in Macmillan is
to deal successfully with a problem or difficult situation
That is not the definition of "manage with" but just of "manage". I can't find "manage with" in either Collins or Cambridge. But I did find it in The Free Dictionary.
manage with (someone or something)
- To be able to operate, function, or carry on adequately with a smaller amount of people or things than one had anticipated. To do as well as possible with only someone or something (less than one had hoped for).
- Terry called in sick at the last minute, so I guess we'll have to manage with just three people tomorrow.
- Commercial airplanes are designed to be able to manage with a single engine if the other one fails.
In the first example, one is basically saying that they will have to manage day to day activities (i.e., function adequately) with just three people. In the second, it says the planes are designed to be able to continue operating (i.e., stay airborne) with a single engine. If you put the first example in the context of "work", it should look like this
Terry called in sick ... I guess we'll have to manage [finishing the report] with just three people ...
And the second should look like this
Commercial airplanes are designed to be able to manage [to stay airborne] with a single engine ...
Yes, there is a difference between the two.
How do you manage those unruly children?
This is asking how you discipline them when they misbehave. It is asking about what action you take, say, when they start fighting with each other, breaking things, running around, etc.
How do you manage with those unruly children?
This is asking how you manage [to do something] with those unruly children [around you]. That "[to do something]" can be anything - work at home, cooking, cleaning, etc.
This is asking how you are able to perform those activities with the given amount/number of "unruly children". And this is where the sentence does not make sense on the surface because, logically, you don't want more of "unruly children".
Again, the definition is "to do as well as possible with only someone or something (less than one had hoped for)." We don't hope for more "unruly children".
But you could think of it this way: You operate or function adequately with the help of a home cook, a babysitter, your spouse, and your three children. However, the babysitter went for a vacation, your home cook called in sick, and your spouse is working overtime. So you had to manage [to do something adequately] with three unruly children.
You had hoped for the cook, the babysitter, and your spouse to be there that day, but you had to do with just the 3 children.