Why do some panels have the annunciator lights and some dont? when are they desired/required and what operations to they commonly represent?
It is a difference in philosophy? In the past airplanes I rented as a fledgling pilot had no or ONE annunciator, typically alternator. You had to scan, Oil Pressure, Oil Temp, Fuel Pressure and electrical output/load. Most of these GA planes are the same today, unchanged (unless they got a Glass panel makeover). You had to scan your gauges. Before discussing "annunciators" in GA lets talk airliners. These have the most advanced "Human Factors" designs. Things changed in the 80's and then again in in the 2000's, still evolving.
Older Gen planes like the B737 had (still do) annunciators galore spread out all over the cockpit panel and overhead, usually near the switches for that system. Because many annunciators were overhead and behind out of your normal scan, to integrate all lights, a master annunciator or "6-Pak" lit annunciator on the glare shield gave you annulation for the 6 systems. (but not spacific) Then you had to look at the panel for that system to see what light was illuminated. Then you had to pull the Non-normal checklist (and be trained to know the name of that checklist for that light which was a talent sometimes). Then you had analyze what was wrong by looking at gauges, or other indication light or lights. It worked great. It was analog and no doubt required lots of wires to this system. It also requires pilot training, knowledge and system understating. One annunciation could mean more than one thing.
Modern jets have EFIS have EICAS which is a master alert, caution and warning system for all aircraft configuration/system status, and that SPELLS out what is wrong in plain language, no interpretation required and guessing what the proper checklist is for that light (which can vary).This is a separate screen for not your PFD or EFIS. (Note you can switch what info is on what screens or compact more info on a screen). It also tells you how critical the issue is (white, yellow, red). More advanced systems give you a schematic of that system and non-normal checklist right on the screen. It will also acknowledge your actions as you progress through the check list, checking off that item. This is way past just annunciations. Needless to say this is way cool. All this information on one screen. There are STILL annunciators but far less and with importance. Needless to say engine fire gets the full Christmas tree, EICAS with appropriate fire handle lighting up and aural warning. The pilot does NOT have to be a system expert, know what checklist to look up. It is all given to you. Cool. Pilots can focus on flying. It's like GPS and magenta line, anyone can fly that. In the day NDB, VOR, LOC, GS, DME required you to mentally get situational awareness, with brain power, not a glass display. Would I go back to the old way? I can and do, but no. The modern EFIS and EICS is superior. Still it does not eliminate the pilot airmanship, skill and judgment.
Now for our GA, Certified, LSA and EAB kit planes. We can install modern EFIS and EIS (engine indicating systems). Most new birds are glass although steam gauges are still offered. GA is not up to airlines, but GA EFIS is way advanced to the rental planes I flew in the 1980's. The power of having all alerts, cautions and warnings on one screen is great. Plus we have way more info. As a plane renter early in my flying career very few planes had even one CHT or EGT, much less GPS. Our engines with glass cockpits have CHT, EGT on each cylinder, FF, with navigation all integrated. What about annunciation? Well it's all on the GLASS.... My Grand Rapids Technology EIS gives me a discrete red light, if anything is out of range it lights. This MASTER warning light (annulation) draws your attention to the EIS. Unlike the B737 of the old days, your attention is to one screen not more annunciators. The data is all on the EFIS. So I have one annunciation, dedicated lightbulb. Without the annunciator light I may not notice the out of range condition, at least as fast. So I am Pro annunciation to supplement the glass.
For cars we have "idiot lights" or annunciators for a long time. However those lights are connected to a computer now. Past cars had actual oil pressure, water temperature and electrical load or charge gauges. As cars have evolved become more complicated, a computer or computers started monitoring all the systems, giving you an annunciation.
YOU can't have enough annunciation lights to do all that.. So you get a light? Go to the manual and it will tell you what is what (often bring to dealer). However what is the value of the exceedance. You have to go to the OBDII reader the pull that data out. Is that better. Well I like to know what the numbers are. There are OBDII readers that will show you the details real time while driving.
I don't hate car annulations. It lets me know when tire pressure is less than 32 psi, low fuel. However the "check engine light" is a fun one. Again you can check for codes. I have very dedicated VW OBD reader that can tell me way more than a generic OBDII reader. I can also program the car. This expensive VW reader has paid for it self many times over. The AC was weak. I was able to read the voltage and current draw of the solenoid for AC variable compressor solenoid. This is not a simple on or off compressor but variable. The solenoid which regulated a swash plate and stroke of compressor was stuck as they do some times. When they do they draw more current. I evacuated the refringent, replaced the valve (with compressor in place), recharged the AC system and it was night and day. That detailed info I was able to figure out what is wrong. There is no annunciator for AC not working well. Now many cars have CAN BUS.... Which is a digital system with remote nodes simplifying the wiring. Planes have this now with electronic electrical systems, with solid state CB's. You can know what every item (or node) is doing and control it digitally. Bottom line annunciation light bulbs are no longer able to keep up with all this digital control and feedback. You need a computer screen. However every super modern jet has some lights. In fact every push to operate switches in jets often have internal illumination and ON or OFF or AUTO annunciation will be exposed (often mechanical shutter). So status of a switch, annunciation at the switch is still a thing. In our case it typically is just switch position (up or down) and a fixed placard.
To answer you.... depends....
(There are people that make a career on subjects like this. With that said I love me some J3 time... no electrical system).