Convert AI picture into vector file with new adjustments

Description

I have made a logo for my business using ai and it is not perfect. I need someone to adjust it to provide me with some variations.Once finished, I would like to have the final version in a vector format so that the resolution remains high quality.
Picture one is the AI photoI would like to take out the 'Che & Co' section And then make the 'London' part larger and higher up a bit.
I will then decide to see if it looks good or not
I wish to add the flower of life pattern to the horse in a subtle way. We might have to edit this and have a few versions of it before I'm satisfied.
I then wish to turn it into a vector file. Although the logo with AI is on a blue background, I would like it to be on a white background also.
Picture two shows what someone else has done but it doesn't look polished enough.