Both answers are right to a point. I used to be a web host and a registered ISP. I was a presenter at the first ISPCon known as ISPOne for USRobotics and represented well over 1 billion dollars in sales in just the first quarter. I have been out of the industry for quite a while, but not too much has changed except for some of the offerings and some of the technology. The newer technologies, however, are based upon older technologies that have been around for 30 years. I will limit my answer to shared hosting only. Here is what you need to know.
Generally speaking, there are as many sites per computer as possible. I know that is a Duh! statement, however, not all hosts do this. Quality hosts gauge performance of their servers and move sites around as needed while others do not care one whit. The common trick in the industry is to put as many sites on a computer and promise more disk space than they actually have. This is because it is rare that anyone uses the all of disk space (as well as CPU and memory) made available to them. Quality hosts will at least monitor performance and allocations and make changes dynamically.
As far as the computers, they are often the cheapest generic clone computers that they can get away with. Quality hosts will use a name brand of course, but use a cheaper model such as Dell over HP. Higher quality hosts will use clusters of computers and SAN technology to allocate resources. There are different cluster based technologies and SAN technology that allow a high number of computers and hard drives to be allocated within a dynamic space and appear as a single entity.
There are usually huge banks of computers so it is not practical that all of them have a public IP address. IP addresses within the hosts LAN is always private IP addresses. The public facing IP addresses are on routers or specialized computers/harware that manage traffic and bandwidth. The public facing hardware will use NAT and/or proxy to direct the traffic to the right computer. It is not uncommon for many thousands of standalone computers to be managed by software and sites moved around from computer to computer seamlessly. As well, it is not uncommon that a cluster based technology and SAN technology is used to host a huge number of sites.
Because of this, there is no correlation between number of domains assigned to an IP address and performance.
Here is what is important:
The hosts reputation. Period.
It does not matter if a bank of standalone computers are used or large-scale clusters with SAN. Obviously the latter is preferred within telecom production environments, however, the difference between the two are really minimal for hosting these days due to options available. For the customer, there should be no difference. What is important is that the host cares about the customer and responds to issues prior to the customer calling. A heads-up monitor, external monitor, internal monitor (per machine), a network monitor should be able to alarm the host immediately when things go wrong or performance is suffering. It should be standard practice if the problem is solved seamlessly immediately before the customer even notices. Fail-over of all stripes, hot spares, OTS (on the shelf) pre-configured hardware spares, spare in the air hardware, snap-shots and images, dynamic host allocation, and fast networks should make moving and recovery very easy and fast. If standard practices are observed, the customer should never have a problem that they did not create and fixing it should be a snap.
On a side note, the claim of 99.999% up-time is a statistical impossibility. It is BS plain and simple. I worked as a consultant where %100 up-time was required with an SLA (service level agreement) and fees paid to the customer for anything including a 1 second outage. These fees started at $10,000. In all the years, a fee has never had to be paid. This is possible with hosting but rarely done except at tier level 1 providers. Otherwise, expect something within the 97%-98% (point something) as being standard.