Having read your blog off and on for over a year, I believe you would be happier with oxy-acetylene welding. I believe that it is more challenging than most other common types of welding. This is not to say that it's extremely difficult, it's just that from what I've read, I believe you would enjoy the challenge of welding with oxy-acetylene more than the others. In addition, the skills that you will be forced to learn with oxy-acetylene welding right off the bat will help you should you decide to transition to other types of welding. Finally, if you have an oxy-acetylene setup, you will certainly be set up for cutting metal as well.
There are limitations to most every type of welding and picking the appropriate system for your needs is important. Therefore it would be wise for you to know the primary limitation of oxy-acetylene welding. Since you are using an open flame to heat the metal, you will be spending a lot more time getting your metal to the appropriate temperature than you would with an AC/DC welder or a MIG welder. This means that there is going to be plenty of time for the surrounding areas to absorb heat from your flame. For welding on a bicycle frame, this is probably not going to be a problem. The tubing will be strong enough to absorb and dissipate the heat without distorting the metal. You will also be able to remove other objects from the bicycle that may be affected by the heat. For example, bearings, seats, brakes, etc.
For the money that you're going to invest in this, I would highly recomend the Victor brand of welding equipment. Victor is the standard for oxy-acetylene welding and cutting. You can go with other brands and do ok, but I recommend them for several reasons:
1. You are going to be learning a new skill. Learning with equipment that has "quirks" is going to make that learning process more difficult.
2. After you develop your skill, producing truly professional results with quirky equipment will also be difficult, if not impossible.
3. You will be able to add to your welding kit easily because Victor stuff is available all over the place. If you start with just a few attachments and find you need more later, it won't be a problem.
4. If you decide to get rid of it, people in the know will be interested in purchasing it from you. People in the know will have no interest in purchasing your no-name kit that probably works terribly.
The second Amazon link you had showed a decent looking setup. I would research whether the equipment that comes with the kit is the same as a full blown professional level kit that you would get from Victor. I am not sure if the small kits are of a different quality than the professional level stuff from Victor. If they are the same, I would say that would be perfect to get started with. Looks like you could even strap it to your back so you could ride your bicycle to go get the tanks re-filled.
I would highly encourage you to pursue this endeavor. I would imagine that if you're watching youtube videos of how to weld with oxy-acetylene you may have some hesitation. The quality of the video is poor. You will not easily be able to find someone to help you in person if you struggle with it. It is something totally outside of your realm of experience. But it's just like anything else. It looks more difficult when you don't have the equipment in front of you than it actually is. Once you have some decent quality equipment you will be able to figure it out between youtube and other websites and some time.
One thing you will need though is some scrap steel to practice on. There is a good chance you will mess up the first bicycle you work on. Just make sure you have the opportunity to practice before working on something that is important to you and you'll be fine. Good luck.