How might I be able to give a visible and meaningful sign of my commitment to her?

She's enormously wealthy? Find a palazzo (or a piano nobile of a palazzo) in Venice for her to buy.

"...by sending sweet, funny, 30-second videos you shoot of yourself on your phone..."

Now you've done it. He's going to send her a picture of his penis.

Clitoris, meet tongue.

Well, I kinda know this from HER perspective. I'm not rich, but I do have money and have been well off for at least 10 years, so I just don't remember what beeing poor feels like or that "end of the month"-feeling.

When I have girlfriends, it's difficult. Money is just one ressource, but not a very important one for me, so I don't mind paying for dinners, concerts - stuff, generally. But it gets awkward for these women, sometimes, because they want to give me something back. What they find hard to understand is, that what I'm giving them is of little value to me. I won't notice on my account that I fork out a few hundred bucks on a dinner. I don't care about that money. But I DO care, a lot, about the way they show affection, help me out with stuff. But I do sometimes end up not taking them out for dinner as often as I'd like, to make it less awkward.

Sentimental:

If you propose, give her your grandmother's engagement ring or something with a story behind it.

Is she adventurous? Plan a romantic camping trip in the middle of nowhere, and spring for the most luxurious picnic you can afford, or try home made if you are good at food prep.

If you're both outdoorsy, it helps a lot.

Manly:

Take initiatives. Surprise her. Don't wait for her to suggest activities, you suggest them.

Fix stuff around the house for her. Help her buy a car, deal with electricians, etc. If that's not what she needs help with, find something else where you can swoop in and be the hero defending his maiden.

Be physical... if you can, physically pick her up. Get high things off of the shelf for her. Open jars.

I'll skip whatever I intended to say, and just refer you to NicoleK's post. As usual, she is right on the mark.

NicoleK said some good stuff. Things my husband does for me that don't look romantic but make me a LOT happier:

removing (or killing) spiders and really nasty bugs
opening jars
getting crap in/out of the crawl space
shlepping stuff (like giant heavy bags of yard debris)

I could do those things, or hire some one to do them, but its really nice that he does them FOR me, just because I ask (or he does it on his own).

I'd rather have some one care to remove the spider-on-steroids from my house than an expensive handbag or piece of jewelry.

Yep... what NicoleK said. Little things that cost nothing but time and/or a little labor often mean more than anything my husband can buy me. Killing the spider before I even know it's there goes a long way in my house.

Also, what David said. It seems cliche, but a good roll in the hay is also a great way to say "I love you" when all the bugs are killed, work is done, dinner is eaten and there's nothing but the two of you.

"My girlfriend of a year is enormously wealthy and very generous. Despite my protestations, she loves buying me nice clothes and other gifts, and appears to expect little or nothing in return except my love."

How awful! No, all kidding aside, I get the issue. Here's an idea: Does the LW cook? Making her a really nice meal might do the trick. If he doesn't cook, he could go over to Miss Alkon's Amazon link, and search on the phrase "bachelor cookbook." I saw more than five dozen possibilities when I did that, and plenty of the offerings were aimed at beginners.

Another suggestion: If she is close to her family, do nice things for them. There's something to be said for your mom telling you, "What a nice man! I llke him!"

This sort of puts to lie the notion that romance is not a "league" sport.
LW hasn't read Ladder Theory or anything else by the PUAs flooding the internet. Women VALUE a man for different things than what men consider to be valuable. In a woman's mind, how he makes her FEEL means far more than what he does... as David D says, "A woman does not FEEL ATTRACTION for a man who makes her THINK. A woman FEELS ATTRACTION for a man who makes her FEEL." It does a lot to explain how prison felons get better attention than run-of-the-mill Nice Guys: even bad attention is better than the bland, safe attention men are trained [by our mommies!] to offer when we're growing up.
I ran into this issue with my last romance, who was a yuppie suburban trophy (ex)wife. I was shoeing her horse. I realized the best thing I could offer was a refuge from the chaos in her life. Her friends thought we were perfect. She dumped me anyway, right before Christmas that year. I saw her last week, and she looked like she was in pain (She won't speak to me).

Neither my husband nor I are wealthy, but we are comfortable. We're a late, and combined household, so we tend to keep our money separate. I make more than he does, by about half again. For gifts, he generally gets me electronics, because he knows I'm somewhat of a luddite. He then helps me figure them out, and programs them for me: think Kindle, iPhone, iPod, Mac, iPad, etc.
I get him experiences: hot air ballooning in the Napa Valley, an ultralight ride over Petaluma, a WWII plane ride, season tickets to the symphony, to the ACT playhouse, catered parties with all his friends, etc.
In thinking about it, he gives me what he likes, and I give him what I like, but it all works out for not an enormous amount of money....
He also takes out the trash, mails my care packages to my son in the military, listens intently, is neat and clean, does the laundry and dishwasher without being asked, kills spiders, and rubs my back without expectation. Just be thoughtful...it will become apparent what to "get" her.

Get savage with her.

One word - Skittles

Chateau Heartiste
http://heartiste.wordpress.com/2009/05/19/be-a-skittles-man/

Material things guaranteed to make me go googly-eyed:
- a love letter: doesn't have to be flowery handwritten poetry. Even a short "I think you're cool" note counts.

- mix tape/ CD: Songs that are meaningful, fun, or romantic. Combine with a love letter as a jewel case liner and that's pure gold.

- A homecooked meal: Even if all you can manage is Macaroni and Cheese with hotdogs - put it on a plate and light some candles.

- Anything that says you pay attention to her hobbies, collections, or likes: A 25 cent plastic frog from a gumball machine to go with her frog collection can go over better than a pair of earrings that aren't her style at all.

- Flowers (duh): Even a handful of daisies or wildflowers count

All of these things do exactly what Amy says. They show thoughtfulness and a genuine "I like you enough to be interested in your life."

First-world problem reappears: got rug-burns, wrestling Christie Brinkley.

Count your blessings, Mr. Kerry. The hot chick in the parking lot at Safeway with the Focus, stoner friends and HPV could have been yours, but you seem to have a gift here.