Please help me figure out what killed my betta

I'm sorry about your fish. However, there are a lot of different reasons why fish get sick. Your post doesn't give us a lot of information to go on, but here's what I can determine from it:

Dropsy is a symptom, not a disease in itself. The fluid buildup occurs because the internal organs (especially the kidneys) have begun to fail. The actual disease can be a bacterial disease, a parasite, or many other things. By the time dropsy is seen though, the problem is often severe and the fish's chances of survival are low.

As for your second fish, did you notice any symptoms? For example, was he bloated, or was there a fuzzy substance on him, etc?

If you want people to speculate on the cause of their deaths, you could fill out the information on this sticky. This will give us more details about what happened.

At this point, it sounds like your tank is cycled, so I would do a large water change (about 75%). This will remove any bacteria, wastes, etc from your tank. In the meantime, people will read the information you provide, and offer their thoughts and comments.

Housing
What size is your tank? 10 gallon
What temperature is your tank? 78 degrees
Does your tank have a filter? Yes
Does your tank have an air stone or other type of aeration? No
Is your tank heated? Yes
What tank mates does your betta fish live with? None

Food
What type of food do you feed your betta fish? BettaMin by Tetra
How often do you feed your betta fish? Twice a day, fasting on Sundays.

Maintenance
How often do you perform a water change? Every Wednesday
What percentage of the water do you change when you perform a water change? 25%
What type of additives do you add to the water when you perform a water change? Prime and Replenish for R/O water

Water Parameters:
Have you tested your water? If so, what are the following parameters?

Ammonia:.00
Nitrite:.00
Nitrate:.00
pH:6.8

Symptoms and Treatment
I will do this twice, once for the first betta I lost, once for the second betta I lost:
Fish One:
How has your betta fish's appearance changed? His appearance had started to change, his belly swelled, and his fins would clamp. He would also dart around the tank.
How has your betta fish's behavior changed? He either sat at the bottom, or at the top.
When did you start noticing the symptoms? I started noticing the symptoms about 4 weeks before he died.
Have you started treating your fish? If so, how? I started treating him with tetracyclin because I believed it was a bacterial infection.
Does your fish have any history of being ill? This fish seems to be constantly sick, even though my water was always good. I think I bought a sickie.
How old is your fish (approximately)? I had him for about 8 months before he died
Fish Two:
How has your betta fish's appearance changed? His appearance didn't change at all. He was not bloated, or fuzzy.
How has your betta fish's behavior changed? He was fine in the morning, swimming around like usual, sitting at the bottom of the tank around noon with his fins clamped, and dead by morning.
When did you start noticing the symptoms? Around noon
Have you started treating your fish? If so, how? I didn't get the chance!
Does your fish have any history of being ill? He seemed so healthy when I got him, at least, healthier than my first betta. He swam around ALL the time.
How old is your fish (approximately)? I got him about 3 weeks ago.

Again, I'm sorry for your losses.

It sounds to me like your first fish may have had internal parasites. A combination of metronidazole/praziquantel will treat the most common parasites. (The combination can be found in products such as API General Cure.)

I'll speculate that your second fish either had a fast-moving bacterial infection, or there was a contaminant or toxin in the tank, however I can't say for sure this was the cause of his death.

Your overall care looks good, but I'll make two suggestions:

(1) Switch to a quality protein-based pellet.

Foods that contain a lot of grains/fillers can cause constipation, bloating and/or buoyancy issues in some bettas. Tetra BettaMin's list of ingredients (with all grains/fillers underlined) is: "Fish meal, ground brown rice, torula dried yeast, shrimp meal, wheat gluten, dried potato products, dehulled soybean meal, fish oil, soybean oil, algae meal, sorbitol, lecithin, l-ascorbyl-2-polyphosphate (vitamin C). Artificial colors including red 3. Ethoxyquin as a preservative."

On this forum, two foods are highly recommended. They contain quality proteins, and seem to be well-tolerated by Bettas:
1) New Life Spectrum Betta pellets
2) Omega One Betta Buffet

(2) My second suggestion is that you increase the percentage of your water changes, or do an occasional larger water change.

Even though your ammonia, nitrate and nitrite readings are EXCELLENT, dissolved organic compounds (DOCs) and other compounds can build up over time. The best way to remove them is via water changes.

Many people opt to do one 50% water change per week. However, doing even an occasional large water change will help remove these DOCs and other compounds.

Good luck. I hope that #3 is a lucky number for you!