Generally speaking, cruise missiles are harder to detect and easier to shoot down and ballistic missiles, the other-way around. However, cruise missiles can also be easily detected depending on the situation. Ballistic missiles have can also hard to detect in the sense the enemy may not have enough time to respond to them once detected.
Clearly you do not understand what a quasi ballistic missile is. Those missiles follow a different trajectory than a purely and classic ballistic system. Moreover these systems perform a great deal of inflight manoeuvring. Your imagination of a modern ballistic missile appears to be a system that follow a completely predictable and purely ballistic trajectory, this is an amateur understanding of the situation. The reality is however, there are many different types of systems available today and these can categorised in different ways. The article below illustrates the different categories quite well:
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Even assuming the above was correct in all situations at all times, which it is not, the point is not whether you can see it, but how will you shoot it down?
No, it is not. Cruise missile are easier to procure than ballistic missiles, hence why you see them in more quantity worldwide. Moreover, it vastly easier to create a more accurate cruise missile than ballistic missiles.
You are comparing apples and oranges. A system being created in the US is far more expensive than one being so in Iran. This is all basic economics.
That is not how this works. It is not range that determines price, but the way the system is created. For example, an Iranian Raad-500 with 500km range costs half as much as a 300km ranged Fateh-110. Why? because of more efficient mass production, new materials etc.
Patriots have been heavily upgraded with ballistic missile in mind and they are supposed to be used against such missiles at a different layer compared to THAAD. Thus it is not a matter of whether they were "specifically" designed for it or not but what capability are they claimed to have against missiles. If we take America's own words, then:
The PAC-3 is another upgrade of the Patriot air defense missile system optimized to counther the threat posed by ballistic missiles. It was developed by Raytheon. The Patriot PAC-3 is in sevice with the US Army and some US allies.
www.military-today.com
So I am afraid you're resorting (and failing) to mental gymnastics.