Difference between -게 됐어요 vs. -게 되었어요
I have recently learned the -게 되다 structure, but depending on the source, they either teach forms for all three tenses ( -게 돼요, -게 됐어요, -게 될 거예요) or use -게 되었어요 in every possible sentence without changing the tense. What is the difference between those two? What is the more natural way to use it?

Question

Difference between -게 됐어요  vs. -게 되었어요
I have recently learned the -게 되다 structure, but depending on the source, they either teach forms for all three tenses ( -게 돼요, -게 됐어요, -게 될 거예요) or use -게 되었어요 in every possible sentence without changing the tense. What is the difference between those two? What is the more natural way to use it?

Answer

됐다 is just a contraction of 되었다, so there is no difference in meaning or in any other aspect.  Both are natural, but 됐어요 might be preferred in speech because it's shorter.

Perhaps one of your sources uses only -게 되었어요 to highlight that the past tense form (되었어요, 됐어요) has a special usage.  While all tenses are possible, the past tense (which is also a perfect tense) denotes a case of coming to a certain state, arrangement, or practice,  which can have a somewhat different connotations than their present and future usage, and which can come particularly handy in some real life situations.
- 이번에 결혼하게 되었어요 = I am getting married shortly (i.e. everything is decided and set to go).
- 회사를 그만두게 됐어요 = I'm quitting the company (I have come to this decision for my own reasons).
- 한국 친구와 친하게 지내다가 한국어를 배우게 됐어요 = I came to learn Korean while associating closely with a Korean friend.