welcome to EL&U. It depends on the context like so much in English. If you are writing about the City of London (that's the square mile around St Paul's cathedral, not the rest of it), and particularly about the financial market based there then you should use a capital letter because the word City is then a proper noun. It's a little like calling you Steve or me Ben, it's a short form of the name.
In the UK at least people also use The City as a generic term for the financal market as a whole, for example "Utility privatisation was an opportunity for people to make huge profits in The City." or "City traders made huge profits from utility privatisation." Most of the transactions took place in the City of London but some may have been conducted elsewhere. When you use The City or City in this sense you should alwsys use a capital letter.
The same thing applies when you write about city administrations. For example "In Philadelphia trash collections are the responsibility of the City." However if you were writing about Philadelphia you could also say "The city has many fine historic buildings, some of which are National Historic Landmarks". Here you would not use a capital letter because not all the historic buildings are owned by the city authorities so you are talking about the city as a whole.
So in terms of your question it depends on what is being written about, if you are writing about technological advances in British financial trading you would write "...spark innovation in The City", if you were writing about forces driving the adoption of new technologies by individual entrepeneurs in Philadelphia then you would write "...spark innovation in the city" since neither the London financial markets nor the Philadelphia city authorities would be directly involved. The same considerations apply to your other quotes.
I hope this helps.