Native speaker here!! I think the difference between the two 하나 and 일 is that 일 is from the sound of the Chinese character (一，二) one and two. 하나is a Korean word that's not made up with Chinese character.
You are correct. The Korean vocabularies you are referring to are used by native Koreans. However, you can use Sino-Korean number forms to communicate since native Koreans can also understand them.
Could someone explain the difference between the two Korean number systems. 한 vs 일?
The Korean language has a lot of words that come from older Sino vocabulary and have a corresponding Chinese character (한자/漢字), but also has a lot of Pure Korean words that mean the same thing but do not have a corresponding Chinese character, since they are etymologically unrelated to old Chinese.
Sino numbers and Native numbers are used to count different things. Native numbers count hours, but Sino numbers count minutes and seconds. Sino numbers count years, but Native Korean numbers count animals, cars, trees, etc.
A lot of it is trial and error when you start out, and a bit of memorization for which is used to count what.
한국어는 중국 한자의 영향을 많이 받았습니다. 한자는 음(sound)과 뜻(meaning)을 동시에 가지고 있는데 한글(한국어)은 이를 따로 따로 나눕니다 그래서 숫자를 한: 일(一), 두(둘): 이(二), 석(셋): 삼(三) 등으로 표현을 합니다. 영어로는 one, two, three 는 한글로 하나, 둘, 셋 또는 일, 이, 삼 같습니다. 다만 순서를 나타내는 첫번째, 두번째, 세번째는 first, second, third 로 표현합니다
이를 한글로 기수와 서수라고 합니다.
just wait until you realize there are 4 different 공's 7 if you count suffixes and prefixes.