Well your "what is influence" section is correct. What you're referring to as the influence score is usually called reputation. It is an accumulated score based on your upvotes minus your downvotes (flags). But a person who invests money and only curates would have influence (more SP they bought) but stay at the starting 25 Reputation. Likewise powering down SP would drop your influence but not Reputation.
If you wanted to add in the effects of Reputation, it currently just related to post visibility on Steemit.com and as a badge of honor of activity and goodwill in the community. If a person has a negative reputation their content will be greyed out and given less visual attention, although it is not censored.
The difficulty of moving up reputation levels scales up greatly. Everyone starts at 25 and can be at 45 in no time. The higher rep scores take a lot to reach.