Leaflet.draw mapping: How to initiate the draw function without toolbar?

This simple code works for me:

new L.Draw.Polyline(map, drawControl.options.polyline).enable();

Just put it into the onclick handler of your custom button (or wherever you want).

The variables map and drawControl are references to your leaflet map and draw control.

Diving into the source code (leaflet.draw-src.js) you can find the functions to draw the other elements and to edit or delete them.

new L.Draw.Polygon(map, drawControl.options.polygon).enable()
new L.Draw.Rectangle(map, drawControl.options.rectangle).enable()
new L.Draw.Circle(map, drawControl.options.circle).enable()
new L.Draw.Marker(map, drawControl.options.marker).enable()

new L.EditToolbar.Edit(map, {
                featureGroup: drawControl.options.featureGroup,
                selectedPathOptions: drawControl.options.edit.selectedPathOptions
            })
new L.EditToolbar.Delete(map, {
                featureGroup: drawControl.options.featureGroup
            })

I hope this will be useful for you too.

EDIT:
The L.EditToolbar.Edit and L.EditToolbar.Delete classes expose the following useful methods:

• enable(): to start edit/delete mode
• disable(): to return to standard mode
• save(): to save changes (it fires draw:edited / draw:deleted events)
• revertLayers(): to undo changes