Hi ST (and of course anyone else reading this thread),
"What's more important; what you hear or what the audience hears?"
Interesting question, and not one that I can answer in a couple of sentences. This could take some time so I'll start by commenting on your original post.
Here are some things I've heard from performers.
I have to be inspired by my sound before I can give an inspired show to the audience.
I don't see it quite this way. Having started performing live at the end of the 60's (that's 50 years now), there was really no good live sound to be had anywhere. This would mean that I myself and everyone else (including top professional musicians) could never give an inspired performance in the early days of PA development. There were some really good performances at this time which have inspired (and continue to inspire) musicians for 50 and more years.
To hear oneself well when performing is definitely a bonus and can maybe help to lift the performance to a higher level. Vocal intonation for example is easier when the vocalist can hear his/her voice well and not have to put a finger/hand over one ear to hear the voice through the bones of the head. Most of us have seen vocalists doing this on really loud stages (especially before the use of in-ear monitoring).
Most of the time the audience will not hear the difference between a $300 guitar and a $3000 guitar. So why spend the money?
It's good enough for live
This is probably very true. Most of them won't. If we look at acoustic guitars at this point the pick-up system sounds anything but like the guitar it's amplifying.
Why spend the money?
I can only speak for myself here, but this is how I see it, and it's also how I came to buy my main live acoustic guitar.
If I go on stage playing a guitar that I only play on stage, this instrument is going to feel at the very least a little strange. What I really needed was an instrument that wasn't going to break the bank but also one that I play in my living room.
I kept away from the Gibsons/Martins/Taylors etc. as I would also be paying for the name to some extent. Instead I looked at the top range of less high-end manufacturers and ended up buying the top of the range Seagull Artist Studio CW (cutaway) and had the newly released LR-Baggs Anthem installed. The guitar is so good that it soon became my go to guitar at home and also live. The anthem gives it a really good live sound, and purely acoustically it sounds really good and has "aged" well through being played very often. On stage I have a guitar that is really familiar in my hands because I play it a lot at home.
I would say that this instrument inspires me to play well, and has no hidden pitfalls when I use it live.
I regularly play at a venue with a mediocre sound system. The monitors are ugly, and the sound in the house is muddy. Whenever I play there, I bring my Bose gear even though I know the regulars are happy (enough) with the house system.
I know the feeling. Especially in loud locations (pubs, bars etc.) where people are there to have a good time and the overall sound isn't too important. I would also probably take a compact for my own back-line. It does make me feel better if I have a good sound myself. I suppose that I could say I'm spoiled nowadays in comparison to the early days of my musical "career".
I dropped in there to catch the last set of an up and coming band. I've heard them there before, and the sound was typical for that room. On this night the band was on fire and so was the audience.
The sound was atrocious, much worse than usual. Everything was distorted; vocals, keys, e-drums.
I know this feeling too. Generally there's a fair amount of help coming from the alcohol being consumed. A certain amount of inebriation can go a long way to "improving" one's perception of sound. If it's loud as well, the ears aren't as sensitive as they would be by less volume.
I waited until the show was over and went to visit my buddies on stage. Here's what had happened. The band was uninspired by the house system on previous occasions, so they brought their own board and monitors. They ran two lines to the house mixer plugged into XLR inputs. They didn't set the inputs to Line Level.
Okay - so you have figured out what was wrong.
I guess the inputs were overloaded and distorting pretty heavily. The band probably had OK sound as they were using the monitors they're used to.
Here's what was right. The band had a ball. The audience had a ball.
That's the main thing at such gigs. Not my cup of tea, and probably not a gig that I'd really want to spend a long time at. I can remember seeing the group YES when they were doing the "Fragile" tour 1971/72. The one thing I really remember about that gig (apart from the fact that the band was awesome) was that I'd never heard such a good sound before. It was in a completely different galaxy to everything I'd heard up to then.
What's more important; what you hear or what the audience hears?
I would say that both are important. I know from experience that you'll generally find musicians in the audience at a lot of gigs. They'll be happier if your sound is good.
Also, having a good sound is really good advertising for musicians/bands. If your sound is good in a small (loud) location and you happen to have someone in the audience who promotes gigs, he/she is more likely to be impressed and want to hire you than if your sound is really bad.
So now down to my own thoughts for my own gigs.
I prefer to play more intimate locations if I'm doing solo gigs. they fit my repertoire and performance better. Solo I try not to do open air any more. This means that the audience generally aren't as loud because they're there to listen to my performance. Also, using L1's they don't have to shout if they want to say something to the people next to them as the volume of the system doesn't need to be deafening to reach the people at the back of the room.
If my sound wasn't important to me I wouldn't be spending my money on Bose gear, but would be using some of the cheaper stuff which is on offer.
Using L1 systems means that I more or less get to hear what my audience hears. If a compromise has to be made sound wise (because there's maybe hardly any room for me to set up and I have to stand really near to the system) I'll always try to make the compromise affect what I hear more than it does to what my audience hears. I'm there for them and not the other way round.
For example, I've even been slightly behind and very close to my Compact when playing. It meant that my own sound wasn't as "pristine" as it could normally be. My audience however had good sound.
I think that I've probably written enough now for you to get the gist of my views on the points you raised in your question. I hope it's been of some help and maybe even of some interest.