Outside the context of astronomy "visual observing" could mean just about anything. It also seems redundant; to most people, the word "observing" implies using your eyes. But to professional astronomers, who only study celestial objects with instruments and never with their own eyes, "visual" refers to the tiny slice of wavelengths wedged between infrared and ultraviolet.
As for "amateur astronomy," it definitely includes astrophotography, which is outside my bailiwick. In addition, many people think that the phrase implies somebody who's advancing the science of astronomy. And though that's certainly an admirable goal, it's not one shared by most amateur astronomers — or whatever you want to call us.
By contrast, the word "stargazer" has a whiff of frivolity to it. To check this, I looked the word up in the Oxford English Dictionary, where the oldest listed meaning is:
"One who gazes at stars. Often used as a familiar or contemptuous substitute for astrologer or astronomer."
On the other hand, the American Heritage Dictionary, comparing the words gaze, stare, gape, gawk, glare, and peer, notes that "Gaze is often indicative of wonder, fascination, awe, or admiration," citing "gazing at the stars" as an example of this meaning. That's more like it!
Still, whether you call me a stargawker, stargazer, or stargeezer, the word "gazing" doesn't really convey my relationship with the stars. — it's too passive. Wonder, fascination, and awe are certainly part of the experience, but if that was all of it, I'd get bored in a hurry. I don't just admire celestial objects; I study them. Not to advance the state of the science, not to make the world a better place, but to make me a better person — more perceptive, more aware, more engaged with the world. Sometimes that seems like a completely selfish goal; sometimes exactly the opposite. Either way, I'm stuck with it. It's what I do and who I am.
How about you? Does the word "stargazing" capture the essence of what you do? Can you think of a better word? Does stargazing have a meaning or purpose for you, or is it just a simple, uncomplicated pleasure, like eating a good meal when you're hungry?