The only problem I have with that stance, Rosa, is that the primary feature of the Alarm Ward is that it is unobtrusive. We already know that Wards alert the creator under at least some circumstances. There is proof positive that they alert when they are attacked, at the very least. There are actually two instances of other Ward types referring to Alarm Wards: Obfuscating and Trap. Both of these types say that they "function exactly like an Alarm Ward" or are "similar to the Alarm Ward" and in both cases they are referring to the fact that they don't impede people, they don't function as barriers like a normal ward does.
A Trap ward starts out in an "off" state and then becomes a solid barrier when someone enters into it. And an Obfuscating Ward only blocks Detection spells specifically.
To use a real-life security system example, I imagine normal Wards are like Alarming your doors/windows. They are a physical impedance, blocking entry and unless you are an allowed user, they alert the authorities if you go through them. They always set the alarm off when attacked. On the other hand, Alarm Wards are like motion detectors or invisible trip-wires. They don't block your progress, but set off the alarm if you enter.
This has been a debate on if pressing through wards circumvents setting off the alert. Does anyone think that accidentally walking through shouldn't alert the ward's creator? That is to say, if someone with an active spell isn't paying attention to wards and accidentally walks through one, (which causes an astral intersection) and their spell gets stripped off by the barrier. I think that would be alerted to the ward's creator.