Example 1 I can only speculate on, but the figure given is almost certainly not for the specific flight you're looking up, nor does it correspond to the number of free (or taken) seats on that plane, since airlines do not release this information to anybody. (Exact load factors on individual routes are closely-held trade secrets.) I'd venture a guess that the "22 people currently looking" just means that 22 people have searched flights from Newcastle to Amsterdam, for any date, some time in the last 24 hours (or whatever window is "currently" in their book). Whatever the case, this is indeed just a ploy to make you book faster, and you can pretty much ignore it.
The figures in Example 2, on the other hand, are almost certainly true. Tickets available on a plane are divided into coded fare buckets, meaning they'll (for example) sell ten seats at $99 as a "Q" fare, twenty seats at $199 as a "B" fare, etc. Reservation systems have access to this information, including how many seats are left in each bucket (availability), and you can even look these up yourself by subscribing to a tool like KVS. See How to know a flight is not full? for some more discussion about this. So if the website is claiming that there are only 3 seats left at a given price, they're not kidding.
The main caveat is that the buckets aren't static: there's nothing stopping an airline from deciding tomorrow that B's aren't selling at $199, so let's shift ten seats over to Q at $99. However, the closer you get to the flight date and the more the flight fills up, the less likely it is that you'll get a discount. The golden rule of flights is that if you know you're going, and you can afford the price now, you should book it now.