What are the different kinds of IPOs?

I recently tried to witness an Initial Public Offering on a trading platform. What struck me is that it seemed to be different according to who the investors were. Indeed, at the time some newspaper already wrote an article about how successful the IPO was for the company, people were already complaining that the stock was still not available on the trading platform social community. The day after somebody wrote a long comment saying that:

The process that has been used to launch the IPO (…) is called
"hybrid auction".

1. Some groups of people were allowed to "reserve" stocks at a lower,
   fixed price. These included existing investors, employees, and in the
   case of the company even some apartment tenants

2. The previous point increased demand and market buzz. Then, bids
   were opened for other funds and institutional investors. This
   increased demand and price even more.

3. The stocks entered the market, but first for professional agents
   only. They could still secure a reasonable price, yet this increased
   demand and evaluations further.

4. Finally, the stocks were made available on portals like ours, when
   the price was already sky-high.

It leaves me the feeling of market manipulation as it times a product scarcity, but that’s not my question. My question is: what are the different kinds of IPOs?