Advance tickets for Pompeii

The site you mentioned (this one) is, in fact, the official website for the Pompeii archaeological park, and sadly it's not https...Consider that this site is useful for the info about the park, not for buying the tikets, and on the homepage you can find that:

It is possible to buy the ticket online, through the TicketOne circuit (Official seller). It is also possible to buy the ticket at the entrances, and at the self service ticket offices at Piazza Esedra and at Piazza Anfiteatro

So for enter the park the tikets are mandatory, you can buy them in advance in the self-service tiket offices, at the entrance (probably will cost you less money but more queue time) or online (that will cost you probably more money but less time).
The official seller of the tikets online is, as you mentioned, TiketOne (wich is one of the biggest websites for buy tikets online for events in italy) and it's https. Here is the link to the page about Pompeii: https://www.ticketone.it/en/artist/scavi-pompei/
There you will find two tikets: one for the "main tour" and one for Orione's house and garden (the cheaper one). For some weird reason there are three kind of tikets for the main tour, all for the same price, I couldn't find the difference between the three, I guess there is no difference.