Can I use a personal satellite phone/Internet connection as a passenger in a plane?

Technical Aspects

The key part of the quote about pilots leaving the sat phone on the dash is the 'dash' part. The dash is under a glass surface, which is almost completely transparent to RF signals and has a nice, wide view of the sky, where the satellites are. This will not translate well to the cabin, where the fuselage is not made of glass and is decidedly not transparent to RF signals. And where the only windows are relatively narrow and point sideways, not up. On older aircraft models, the fuselage is made of aluminum, which is quite conductive and, thus, will block your signal. While newer models of aircraft fuselages are often made of lighter carbon composite materials instead, it turns out that they still have to surround it with a conductive mesh for lightning safety, so those probably won't work well, either. Basically, the fuselage acts as a big Faraday cage.

This is why all of the antennas for aircraft systems, including the in-flight Wi-Fi, are located on the outside of the fuselage, typically inside some type of aerodynamic, but RF-transparent fairing mounted to either the bottom or top of the fuselage depending on the sort of network it needs to connect to. Antennas for terrestrial networks will generally be located on the bottom and antennas for satellite networks will generally be located on the top because airlines usually like to operate their aircraft in such a manner that the ground is more-or-less down, while the sky is more-or-less up.

If you're sitting at a window seat and can hold the sat phone's antenna up to the window, then it's possible that you could get a reasonable connection. I've done that with GPS receivers in a cell phone, for example, and could get a usable signal, even despite how weak the GPS signals are. However, if you're in the middle section of a widebody, you can probably forget about it.

Legal Aspects

The legal question can vary dramatically depending on where you are and each jurisdiction often has quirky rules that often don't make a lot of intuitive sense.

For example, in India, the personal or commercial use of a satellite phone is illegal, regardless of whether or not you're on an airplane.

Another quirky example is the United States. The use of cellular radio signals is banned while in flight in the United States. Another odd quirk is that this rule technically applies even to the crew, not just passengers, and even on private aircraft. Oh, and the FCC itself has said that the rule is outdated since at least 2004, but it's still on the books and still enforced with regard to airline passengers (though generally ignored for private pilots.) However, that ban applies solely to cellular radios and so sat phones are perfectly legal.

However, using a sat phone (or anything) to actually make a voice call is another matter. Some airlines (such as Delta) explicitly ban making any sort of voice call during flight simply because most passengers don't want to be disturbed by others making calls during flight and flight attendants don't want to deal with the ensuing conflicts and disruptions. And, yet other airlines (such as United) ban any sort of radio transmitter, except for the specifically approved ones, such as Wi-Fi and Bluetooth.

Until relatively recently, there were several Asian countries (including China and South Korea) that banned the use of any sort of wireless transmission for the duration of flight (including even Wi-Fi and Bluetooth.) I think most of these have been repealed over the last several years, but I wouldn't be surprised if there are some still hanging on out there.

In short, the only general answer for knowing whether you would be allowed to use a sat phone for a given flight would be to check directly with the carrier you intend to fly. And, for international flights, ask specifically about the route you're interested in if the airline doesn't have a general ban on them for all of their flights. And remember that an airline's rules with regard to in-flight conduct typically carry some degree of force of law almost everywhere in the world, so intentionally violating them isn't a good idea. Similarly, even if the airline's rules don't specifically ban something, if a crew member tells you to stop, then stop.