“Okagesama de” is the part that is more likely to be omitted, I believe. The entire sentence means “I’m well, thanks to (everyone and all good influences).”
I think for a foreigner it makes more sense to consider levels of politeness rather than colloquial vs. school-taught textbook speech.
Pretty formal: “Okagesama de, genki desu.”
Less formal: “Hai, genki desu”
And less formal: “Un, genki yo.”
(There are actually a lot more variations than this, but this gives you an idea.)
These are all expressions that might be used by a native Japanese. Which one they use is determined most by the status of the speaker compared to the status of the listener. The first one would probably be said by a woman, especially to a man (not her husband, probably) in a position of honor or authority. The third might be said by that same man to that same woman. The middle one might be said by one male co-worker to another male co-worker who outranks him.
I am not an expert, but I have lived with and around Japanese people for the past 37 years.
eta: “Okagesama de” might be said by itself, but it is not particularly colloquial, it is just assuming the positive answer to the “healthy?” question. About the same level of formality as the first example.
“Okagesama de, genki desu” sounds unnecessarily verbose. You might see it in a textbook but in real life, it will likely be shortened to something like “ee, okagesama de”. Okagesama de means “thanks to you” or possibly “thank god” but it’s OK to say it to someone who isn’t responsible for your well being.
Also note that “genki desu ka?” is not as generic/casual as “how are you?”. I know Japanese textbooks will provide it as the translation for “how are you?” but it’s more like “I trust you are in good health?”. I would say that to my family or friends, but not a colleague I’m not friends with, let alone someone I just met.
I rarely ran into “Okagesama de, genki desu” just out of the blue. It can be used for someone who has recovered from an illness “Okagesama de, genki ni narimashita” or “Okagesama de, taichoo mo yoku narimashita” “Thankfully, (I or they) are better now,”
“Okagesama de” is more frequently used in business. One should know and frequently use this word. It can be used to answer a question on how business is, when things are going well or if a particular matter was successfully dealt with or a deal was signed.
It’s impolite to brag, of course, so if one is going to – gasp – admit that things aren’t terrible, “Okagesama de” allows you [del]place blame[/del] give credit for this good fortune on the gods, the listener or an unnamed someone else. Anyone other than yourself, naturally.
Japanese society is truly concerned about the group, so it’s awkward for one person or company to be doing better than another. One always avoids bragging about one’s own talents, so allowing an unnamed someone else credit allows you to discuss your successes without seeming uncouth.