Why do vertex labels disappear when I edit with drawing tools?

Since this is a problem of the interactive drawing interface, it may be OK to work around it in an interactive way, too. It's a hack, of course, but don't blame me.

Here is a sequence of steps that works. It's based on point-and-click interaction, so I can best describe it visually in steps. For that reason, I made the ImageSize in the original graph smaller:

GraphPlot[{Subscript[δ, 1] -> Subscript[X, 1], Subscript[δ, 2] -> Subscript[X, 2], Subscript[δ, 3] -> Subscript[X, 3], Subscript[δ, 4] -> Subscript[X, 4], Subscript[ξ, 1] -> Subscript[ξ, 2], {Subscript[ξ, 2] -> Subscript[ξ, 1], "\!\(\*SubscriptBox[\(ϕ\), \(21\)]\)"}, {Subscript[ξ, 1] -> Subscript[X, 1], "\!\(\*SubscriptBox[\(γ\), \(11\)]\)"}, {Subscript[ξ, 1] -> Subscript[X, 2], "\!\(\*SubscriptBox[\(γ\), \(21\)]\)"}, {Subscript[ξ, 2] -> Subscript[X, 3], "\!\(\*SubscriptBox[\(γ\), \(32\)]\)"}, {Subscript[ξ, 2] -> Subscript[X, 4], "\!\(\*SubscriptBox[\(γ\), \(42\)]\)"}}, VertexLabeling -> True, DirectedEdges -> True, VertexCoordinateRules -> {Subscript[ξ, 1] -> {0, 3}, Subscript[ξ, 2] -> {0, 0}, Subscript[X, 3] -> {Sqrt[3], 1}, Subscript[X, 4] -> {Sqrt[3], -1}, Subscript[X, 2] -> {Sqrt[3], 2}, Subscript[X, 1] -> {Sqrt[3], 4}, Subscript[δ, 1] -> {Sqrt[3] + 1, 4}, Subscript[δ, 2] -> {Sqrt[3] + 1, 2}, Subscript[δ, 3] -> {Sqrt[3] + 1, 1}, Subscript[δ, 4] -> {Sqrt[3] + 1, -1}}, ImageSize -> 120] 

Now copy the output into a new line, pasting it twice into a Show statement as follows:

Don't evaluate the cell yet. First invoke the graphics editor and modify the left-hand plot by changing the arrow heads in the right-hand plot to your liking.

Still don't evaluate the cell yet. Change the Opacity of the edges in the right-hand plot until they are invisible:

Then evaluate the Show:

So I superimposed two copies, one with the right arrows but no labels, and one with no labels but the right arrows. The latter has to be drawn first so that the labels are shown on top.