There was no single Norse (much less 'Viking') culture. (and I wouldn't recommend that website; get a good book written by a professional historian instead; in fact if they refer to all Viking Age Scandinavians as 'Vikings' then that's as good a sign as any to avoid it) Different regions had different cultures and not least different laws. And under those laws people from different regions had different rights as well. (e.g. killing a Dane or Norwegian is a less serious crime in Äldre Västgötalagen than killing a Swede)

There were some common features though. We also have no legal codes (or any other documents) from the actual Viking Age; the earliest ones are written down around the year 1200, a century and a half after the Viking Age. However at least the older parts of some of these (Gultatingsloven, Äldre Västgötalagen) probably do date back to the Viking Age.

First off, punishments were meted out by a tribunal, the þing (or thing) which were regular local meetings were crimes were tried and decisions made. "Outlawry", also known as "banishment" (Old Norse term friðlaus or biltogher) was a specific punishment, it was not a 'system' or a replacement for executions or tribunals. Except insofar that it was usually the default judgment for refusing to show up at the þing.

That site is plain wrong claiming there "was no formal death penalty for those crimes the Vikings considered worthy of execution". Execution absolutely existed as a punishment. It's in the laws, for instance as a punishment for stealing; for which Västgötalagen prescribes the guilty be sentenced to "ugs ok til hangæ" - decapitation and hanging. (although probably not both; there's an alliteration here) There are also sagas and foreign sources that speak of the execution of criminals, as well as archaeological evidence in the form of executed people. (for instance, two decapitated fellows of the late 10th century found in Falköping town square a few years ago)

Property crimes were more serious than murder. Even stealing someone's crops from their fields could result in a sentence of execution and forfeiting all your property. Speaking of which, loss of property was another part of a sentence (depending on the law and crime). Because you could both lose your personal property but it would be passed on to your family as if you'd died, but in more severe cases the property could be declared forfeited to the king or as restitution to a victim. You could also forfeit your land as a sole punishment.

But killing someone was not necessarily the most serious crime. It was an society with a strong concept of honor and also with a broad spectrum of what a life was worth. Killing your own slave is perfectly legal. Killing someone in a fair fight means paying restitution, but not necessarily any punishment beyond that. If a slave killed a free man then the slave-owner was responsible to pay restitution, and the slave was not to be counted as the bane of the person in question.

Killing someone who was in no position to defend himself was a far more serious offense, and many such crimes are termed niþingsvärk (works of nið) in ÄVgL. For instance, killing someone in their sleep, killing someone at the þing, killing someone by binding them to a tree in the forest, cutting their hands off, cutting their feet off, killing someone in a sauna or bath, killing someone by shooting them through a window, killing a woman and killing someone with poison. Which also demonstrates (as opposed to what that website says) that nið was not a term for sexual depravity, although it was used in some contexts to refer to such acts.

So anyway, the short of it here is that banishment was not some alternative to a justice system. If you caught a thief, you could tie him/her up and bring her to the þing, and a death sentence could then be pronounced and carried out. Rather, banishment was the second-most severe punishment and would remain so for centuries after the Viking Age. (being 'on the books' as a punishment as late as the 17th century)

As for why this was used; first there was no police force or strong government that could set out and find people who'd often have absconded anyway rather than stick around for their sentence. It also appealed to a sense of 'fairness', it was the next-of-kin of the victim that was owed the debt and therefore up to them to claim vengeance if they wanted it. A lot of these laws already assume that some form of 'justice' has been meted out; for instance catching a man sleeping with your wife and then killing him. (in which case you must bring the bloody bed-sheets to the þing and convince the jury of your case). For those without wealth and contacts, it was often a de-facto death sentence anyway; as long-term survival alone was not really doable. You couldn't really go somewhere where nobody knew you either, as strangers were treated with intense suspicion as well; not least because they could be criminals or escaped slaves. But even if they weren't, strangers, foreigners, beggars, players and other itinerants had fewer legal rights and protections.