Re: One network?
It's not who pays that matters, it's who forms the contract.
You form a contract with the cabbie. He agrees to bring you to Victoria Station, and you agree to give him an acceptable form of payment. In this case, a payment made using a VISA card [other cards are available]. This is the contract that decides whether the cabbie is a contractor or an employee. He doesn't require that the payment be by card, nor does he require that the payment be backed by your employer: as long as its an instrument that results in money reaching his bank account, he'll provide the carriage service. And an entertaining commentary on London's rich ethnic diversity (kidding).
In the meantime, there are two other contracts in place. Your your employer has a contract with you, part of which is that they give you a credit card with which you can pay for legitimate business expenses, like taxis.
The second contract is between your employer and VISA, in which VISA agrees to accept and process electronic payments and give the money to the person who asks for it (the cabbie in this one case, but also the hotelier, pret-a-manger and Network Rail at various other times in the day), in exchange for your employer eventually giving VISA back that money, plus a fee for providing the service.
So, yes, the money to pay the cabbie eventually came from your employer, but he didn't negotiate a contract for taking you to the train station with your employer, he did that with you. You used the benefits of a contract you had with someone else (your employer) to fulfil your obligations with the cabbie, but that's the way the world of commerce works: you form contracts to fulfil the obligations of other contracts..
Plus, even if your example were valid, the cabbie would not be an employee of your company unless he only ever drove for payment on that company's credit card.
Uber is different because it is Uber who always pays the driver. You pay Uber, not the driver, and the driver is later paid by Uber for providing the service that you booked through Uber. No contract is ever formed between you and the driver.
My big concern with Uber from a customer viewpoint is the lack of trust and accountability - even with the worst minicab company (and Uber is basically an online minicab company with a huge fleet), you can find the office, and have it out with the owners if you're treated badly or ripped off, and it serious cases, complaints will result in a driver being dropped. Over time, the company builds up a stable cohort of drivers who are honest and safe, or it goes out of business. Uber is hampered in doing this by its size - it has thousands of "drivers" on its roster, with as many bad apples as you'd expect from any service business of that size. But that sheer size, and the transience of the workforce means you're playing whack-a-mole when it comes to removing bad elements, and the risk of harm or crime remains pretty much constant. Very low, granted, but constant.