First is a service offered by most brokers. You tell the broker you want to drip and the broker goes to market and buys as many shares as the drip dollars will buy, then credits those shares to your account.
The above drip does not involve the company and it does not change the number of shares outstanding. It is just a service provided by the broker.
The second is a company sponsored drip. In this kind, the company issues new shares at a formula price, with or without a discount. If your shares are held by a broker (known as street name), the broker must go to the company (or its drip agent) to participate in the company plan.
Here, ETP has a plan whereby unitholders can use their distributions to buy new shares from etp at a 5% discount. They pay an underwriter a discount to underwrite a public offering, so here they are offering us a bigger discount to buy new units.
So, Etrade and Vanguard go to market and pay whatever the market price is to buy drip shares from some other unitholder.
Ameritrade and Fidelity go the company's agent to buy new units at a discount. Here, no other unitholder is selling, the company gets new cash and issues new units. The market is not involved, except as a reference to set the formula price.