The basic problem here is that your question, as stated, is unanswerable. So rather than try to answer it, I'm going to show you why.
The reason it's unanswerable is because you're trying to map abstract mathematical concepts onto real physical systems, and they just don't fit.
So I'm going to rephrase your question in a variety of ways and show you how one can, simply by redefining the mapping from abstract to real, that one can get whatever answer one wants.
Are there any systems in the known universe that can be experimentally proven to be unstable for either the impulsive or BIBO instability cases?
No. Because the universe is finite in both physical extent and energy content. So there are no systems that have or can have infinite extent, energy, luminosity, or whatever. Nothing physical matches the abstract mathematical definitions.
This leads to an interesting and very practical problem -- how do you define "instability" for practical purposes? Engineers don't usually hang out at the water heater discussing this, because for practical purposes if it grows without any bound except that it breaks itself, or puts itself into an unusable operating mode, it's "unstable".
So a motor whose position grows without bound until it smacks into a stop, breaks the stop and destroys the thing its in is part of an "unstable system", even though once the smoke clears and the bits stop bouncing on the floor, everything is bounded in a mathematical sense.
Ditto, an amplifier that has a sinusoidal mode that grows without bound until the amplifier overloads and then the system sits there in a perfectly stable in nonlinear dynamics terms limit cycle is, for practical purposes "an unstable amplifier".
OK, smart@**, limit your answers to open-loop systems.
"Open loop" is a mathematical abstraction. Any physical object in the universe is held together by particle interactions, and these interactions operate by a form of feedback. So if you drill down far enough, there is nothing made of matter that doesn't, at some level, operate in "closed loop". Electrical systems are made of matter, ergo, they all have some sort of feedback.
Even limiting the question to electrical feedback leaves us with a full, practical expression for something mundane like, say, an actual resistor -- an actual resistor has capacitance, inductance, resistance and feedback between these effects (or "elements", if you don't like the word "effects"). Any object that interacts with electrons has the same property, if you drill down far enough. Electrical systems, by definition, interact with electrons, so they have some sort of feedback.
So, for practical purposes, you have to define what you mean by open-loop. If you're designing a control system at the block diagram level and you have a block with a transfer function in it, then at that level of the diagram that block, in itself, is "open loop". It may stand for a physical part that has a dozen intentional feedback loops in it (and is held together physically by another practical infinity of "feedback loops), but for that block diagram it is open loop until you wrap a loop around it.
You linked to two block diagrams that exemplify "open-loop" and "closed-loop". Remember, block diagram designs can be hierarchical. So if I am buying a motor with a built-in controller from you, and all I care about is getting a certain RPM out of it, then my block diagram is "open loop" -- I issue commands, the motor does what it does. On the other hand, your block diagram, as the designer of the motor/controller system, is closed-loop, because your part of the design has feedback (probably multiple feedback paths, in fact). So, when I tell my boss that "the system is open loop" I'm right, in a certain sense -- and when you tell your boss "the system is closed loop" -- you're right.
Which boils down to: the terms "open loop" and "closed loop" only have local meaning. You simply cannot point to a physical system and make a global statement "that's open loop" (although arguably you can always say "that's closed-loop"). You can only say "I can accurately analyse this system as if it were open loop", or "in order to accurately analyze this system I must take into account its feedback".
Yes, but I meant open loop and voltage controlled electrical systems.
What do you allow inside this electrical system? How do you define what is a control, and what is a power connection? Unless you're designing a power supply, your circuit analysis will take the various power supplies as given, not as "inputs".
So in the example you cite, the bias to a transistor isn't a control input, it's a bias supply that's derived from the power supply -- it's just a given.
If you exclude exterior power supplies, what about batteries? Put a battery inside of something, put a latching relay on the interior power line, and an impulse on the "on" lead will turn the thing on, and then whatever is inside will be stable -- or not.
If you define an "electrical system" as something that starts with no stored energy (i.e., no batteries), and whose power feeds are designated as inputs, then by that definition the system must be stable to impulses. Further, for any real system it'll be BIBO stable, first, because see discussion about the finite nature of the universe, above, and second, because capacitors leak.
If you define an "electrical system" as something for which the power inputs are assumed, it may be autonomously unstable, without needing an input at all.
My real point is you must define how you are mapping from the abstract to the real. Moreover, you must define what you mean by "unstable". For each combination of those mappings you get different answers. In the original question you asked two different engineers, and got two different, and opposite, answers. That's because each one of those engineers had a set of mappings in their head, and considered that mapping to be the one and true mapping.