Convert AI picture into vector file with new adjustments

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 photo
I 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.