well water is always fun! often hard to find the problem and then fix it!

The first leak showing, were you driving and how fast? the second time. same or not?

In your picture I can't see if its coming over the dash bumper or not. It looks to be entering between the leather and the walnut dash, yes or no?
Lets start with the lower screen seal. Two possibilities here, the first being a weak spot along the length of the seal or debris trapped under it, typically from washing the car which allows water to access the inner scuttle area. This is then driven in by the wind pressure whilst driving and you'd be surprised at how little access it needs often not allowing you to replicate it with a hose when stationary!. The other place to look is at the outer edges of the screen where the seal does a change of direction as it follows the scuttle curve and here a potential opening can be created allowing water again to enter the inner screen scuttle area. It then tracks down the inner seal to the centre screen area and then finds the heated screen wire holes, there are 2 or these about 2" apart, it then runs inside the short distance to that join area where the scuttle bumper lower side meets the walnut dash. right where your trickle is in fact! This could have been the result from either potential leak area as above.

Second possibility is very much a windscreen washer issue! First time, were you using the screen wash but not the second time or not as much. Don't know how old your car is but Morgan fitted a black rubber washer hose for the washers threaded through and around all the dash electrics! Notoriously it failed as did the fittings/connectors etc. where it fails dictates where it squirts when the washers are used and if its in the washer jet area it could be squirting right onto the scuttle by your tell tale leak. You really need to drop the dash and have a look inside whilst operating the washer system. Don't panic its an easy little job and PM me if you are worried about it , there are one or two little things to know to prevent damage!
If you find clear plastic pipe when you look it probably means yours has been addressed but still check as I've seen poorly fitted replacements made especially if they used the old fragile connectors!

Lastly really are the washer arm spindle seals and the seals to the jets against the outer scuttle its self. Are they loose? I tend to drip ACF-50 around mine using a thin screwdriver as an applicator. If you grab the wiper arm where it screws to the spindle and push it in and pull it out you will see it has some play in and out as you pull an also if you rock it side to side there will be a clearance play too! Here I use the screwdriver again to apply a very small amount of hypoid diff oil, just two little drips and then run the wipers, stop them and one more drip, then run them again. this oil is very thick and tends to stay within the wiper box spindle.. then I wipe petroleum jelly around the join of the inner and outer joint here pushing it into the shaft and I repeat this once a year at service time, no leaks and no seized spindle boxes.

I hope this helps you find the leak! Its definately an upper one, just a case of finding it now!