We might be getting somewhere.
Firstly your green trace ( "trigger" ) is nowhere near as clean as the yellow trace nor is it switching rail to rail. That may be your probes needing adjustment, you may have "AC" selected as thought earlier, or it may be indicative of some wiring fault, a short to some other component.
Although you don't tell your Proton+ code to loop, you don't appear to tell the code to stop either, so the PICmicro is likely to continue executing after 'bstop', run through NOP's and then wrapround to the start of code and begin to execute again, thus actually looping. The other possibility is that the PICmicro is reseting and starting execution again.
Comparing your 18x.doc with 18x2.doc, the same code now appears to be producing a different result, address plus no data this time. That could be significant.
It also appears that you are saying that if there is no loop in the 18X code, the code appears to loop. That suggests that the PICAXE is reseting, because it would not otherwise loop. If the reset is happening after address is sent but before data is sent that could explain why no data is seen.
With respect to Out 0 ( 18x2.doc ) that goes high after some I2C bus execution. That could be caused by some effect of what has been sent, possibly by a reset. If Out 0 were held low, externally pulling it high would produce a high current condition which could damage the pin. In 18x2.doc the rise time of Out 0 is very slow, so it could be that Out 0 is becoming an input as a result of reset and is then quickly floating high being pulled up by an external source. When an actual output, it is pulling the external influencing high signal low. This would be the case if Out 0 had a pull-up to +V or other high signal.
You say that Out 0 then goes low after a few mS. That could be the PICAXE coming out of reset, making Out 0 an output and set low. Out 0 may also become an input for a brief period when the download mode is entered but I am not sure of that.
Yes the PICAXE is an interpeter. The Basic code is converted to tokens not to any .HEX machine code.
As to the problem with the latest editor I cannot help there. Maybe someone else knows a trick to get a messagebox to appear when a top-most window stops and has lost focus ?
With what you've given us here, my suspicion is that the PICAXE is either going into reset or entering its download mode. Ensure that you do have a 4K7 pull-up to +V from Leg 4 and that the 10K/22K download interface is connected to Leg 3 or that leg is pulled to 0V via a 10K. I would recommend uninstalling the latest Programming Editor and re-installing the previous version which did work, and have a look at both Leg 3 and Leg 4 with your scope to see if there is any activity. After powering up, during normal operation, Leg 3 should always be 0V, Leg 4 should always be +V.
I would also recommend that you check or adjust your scope or settings so that both probes give the same readings from the same signal. A short PICAXE program which toggles an I/O line should give you a nice, clean 0V/+V square wave to work with.
As to why behaviour seems less than random as might be expected for reseting, this could be "Sod's Law" ( nothing behaves as it would be expected to ) or the I2C transmission has a subtle effect which is enough to provoke a reset.