How can I create named edge "types" in Graphviz/dot/neato?

I need to draw a diagram with graphviz/dot where there are common edge types between nodes and am trying to find a way to define a label for each type of edge and then use that label multiple times in the diagram.

For example imagine the traditional ceiling fan FSM example where it's initially in state OFF and every time someone pulls the cord it changes to a new state based on the speed of the fan:

     Pull         Pull        Pull
OFF ------> HIGH ------> MED ------> LOW
 ^                                    |
 |                Pull                |
 +------------------------------------+

Every edge is named "Pull" and I can define that in dot by using:

digraph fan {
    OFF  -> HIGH [label="Pull"];
    HIGH -> MED  [label="Pull"];
    MED  -> LOW  [label="Pull"];
    LOW  -> OFF  [label="Pull"];
}

BUT I don't want to keep specifying the same textual label every time because

1. My labels can get quite long so that's error-prone, and
2. My edges have other attributes like color in addition to label, and
3. I have a selection of multiple different types of edge so I want to make SURE that edge type "A" used in different contexts in the diagram always has all the same attributes.

I expected dot to have a syntax that would let me define names for my edge types, something like:

digraph fan {
    edge_a [label="Pull"];

    OFF  -> HIGH edge_a;
    HIGH -> MED  edge_a;
    MED  -> LOW  edge_a;
    LOW  -> OFF  edge_a;
}

but of course what the really does is create a node named "Pull" and unlabeled edges.

I've been searching online for a few hours with no success. Anyone know how to define edge types up front to be used in multiple locations?

Update: @vaettchen had suggested defining an edge type then listing all of the transitions for that edge type, then defining the next edge type followed by it's transitions. While that would technically solve my problem, it would introduce a couple of others because my graphs today can look like:

digraph {
    subgraph cluster_1 {
        a -> b [label="type x", color=red, style=solid];
        b -> a [label="type y", color=green, style=dashed];

        b -> c [label="type x", color=red, style=solid];
        c -> b [label="type y", color=green, style=dashed];

        c -> d [label="type z", color=blue, style=dotted];
    }

    subgraph cluster_2 {
        d -> e [label="type x", color=red, style=solid];
        e -> d [label="type y", color=green, style=dashed];

        e -> f [label="type x", color=red, style=solid];
        f -> e [label="type y", color=green, style=dashed];

        f -> c [label="type z", color=blue, style=dotted];
    }
}

and to rearrange that by edge type I'd lose the immediate visual clarity in the code of having the bidirectional edges next to each other (a->b and b->a) and I'd have to explicitly list the nodes within each subgraph and I'd have to pull the subgraph-internal edge definitions up into the main graph:

digraph {
    edge [label="type x", color=red, style=solid];
    a -> b;
    b -> c;
    d -> e;
    e -> f;

    edge [label="type y", color=green, style=dashed];
    b -> a;
    c -> b;
    e -> d;
    f -> e;

    edge [label="type z", color=blue, style=dotted];
    c -> d;
    f -> c;

    subgraph cluster_1 {
        a; b; c;
    }

    subgraph cluster_2 {
        d; e; f;
    }
}

So while it would solve the problem I asked about and I appreciate the suggestion, I'm not sure it's worth the tradeoff as you end up with the equivalent of a C program where you had to define all of your variables outside of the functions and organize them by their type rather than logical associations.

To be clear, given the above example what I was really hoping for would look like the following if such an "edge_type" definition keyword existed:

digraph {
    edge_type edge_x [label="type x", color=red, style=solid];
    edge_type edge_y [label="type y", color=green, style=dashed];
    edge_type edge_z [label="type z", color=blue, style=dotted];

    subgraph cluster_1 {
        a -> b edge_x;
        b -> a edge_y;

        b -> c edge_x;
        c -> b edge_y;

        c -> d edge_z;
    }

    subgraph cluster_2 {
        d -> e edge_x;
        e -> d edge_y;

        e -> f edge_x;
        f -> e edge_y;

        f -> c edge_z;
    }
}

Not really an answer but "food for thought" as I don't think named labels exist in graphviz: You could define a default label for the following edges. That works well if your workflow allows to define edges all in one place. Example:

digraph rs
{
    node[ shape = box, style = rounded]

    edge[ label = "pull" ];
    { A B } -> C;
    G -> H;
    C -> D[ label = "stop" ];
    edge[ label = "push"];
    D -> { E F };
    edge[ color = red, fontcolor = red ];
    { E F } -> G;
}

which yields

I have also tried to implement your diagramm with

digraph fan 
{
    splines = ortho;
    node [ shape=box ]

    edge [ xlabel = "Pull", minlen = 4 ];
    { rank = same; OFF  -> HIGH -> LOW; }
    LOW:s -> OFF:s;
}

which produces

so it looks good but with all the tweaking is difficult to expand.