Both of the sentences from the dictionary are correct. Most natives would not even notice the difference between them that you've observed. You can say that the channel has backed up or that the contents have backed up; the meaning is the same.*
Both of your revisions are correct, too. However, normally we would use the passive voice for the road:
The road was backed up for miles because of the accident.
I think this is partly due to the channel/contents distinction, but I don't think that would be a very consistent guide to whether to use active or passive voice.
It will help to understand the basis of this sense of backed up, since it's a secondary sense, though explanations of this kind of thing are necessarily hazy. The primary meaning of back up is to move backward. In the case of flow into a blocked channel, the contents accumulate as they flow into the blockage. Thus the earliest point along the channel where flow is blocked "moves" backward as more of the contents accumulate. This is the main reason why back up makes sense as a name for what happens at a blocked channel.
The point along the channel that is "backing up" as new contents flow into the blockage is more naturally associated with the contents than the channel. The channel is unchanging while the contents flow in and accumulate. So, the contents make sense as the subject of back up, although the "back-up" is easily extended to name the whole situation.
The subject as the cause (agent)
However, there are other, stronger pressures in English at work here. The subject of a verb is often the agent that causes the action, not necessarily the thing that changes during that action. For example:
The kitchen maid poured the water out of the jug. (The jug flowed; the kitchen maid caused it.)
The plumber unclogged the drain. (The drain's state changed; the plumber caused it.)
You could even say:
The accident backed the road up for miles.
Subject shifting between agent and patient
English frequently allows the subjects of verbs to shift between these senses of the thing that undergoes the action and the thing that causes the action, like this:
The chicken is cooking on the stove.
The kitchen maid is cooking the chicken.
The fat from the chicken clogged up the drain.
The drain clogged up due to the fat from the chicken.
Another factor in back up is that it evokes up in the sense of something going wrong, by (weak) analogy with phrasal verbs like messed up, screwed up, etc. or (more strongly) clogged up, choked up, etc.
Yet another common role for the subject is to denote the thing with something wrong with it, when the verb says that something is wrong. In accordance with the subject being the agent, the subject also often denotes who or what is to blame.
The quarterback lost us the game. (The quarterback is at fault.)
The drain backed up. (The drain failed to do its job. There's something wrong with the drain.)
Now, at last, hopefully, you can see why the road normally gets the passive voice when it's "backed up": the back-up is not the fault of the road. The passive voice makes the road a sort of innocent victim of the accident. The traffic is also more naturally seen as the agent of blocked flow, since traffic moves and is driven by people while the road just sits there passively.
On the other hand, if you were arguing that a road is too narrow and needs to be widened, it would make sense to choose the active voice in order to blame the road:
That road backs up for two miles every day at 8:00 and 5:00 when people are going to and from work.
It has less to do with channels vs. contents than with notions of agency and blame—but the peculiarities of channels and contents do figure into the way they get sorted into categories of agent, patient, object with or without flaw, etc.
* I'm saying "channel" where you said "facility", and "contents" where you said "user". There is no standard terminology here, but "channel" and "contents" follow the ordinary meanings of these words. "User" suggests a human being, and "facility" is very broad and consequently doesn't suggest your intended meaning as clearly as "channel".