Rare vs Common diseases??

That would depend on the frequency of symptoms associated with the rare illness. Assume a person comes into the doctor's office reporting that their hair is turning purple. An unusual symptom like this would likely prompt the need for blood tests to determine the underlying cause. The physician may also review medical literature, conduct additional tests, or collaborate with other medical professionals who have experience with similar cases, since "purple hair" may be associated with a previously unidentified medical condition.

Again, its "rarity" status depends on how many people have reported the same condition; I would assume if the problem is large enough, subjects would be genetically tested to determine if there is a genetic component to the disease too.

But lets say something like ALS or throat cancer are considered rare, but how few people having these diseases at the same time would make it rare?

If you review the cause of death of let's say the last 10,000 people that died, I'd assume that very few of them would have been diagnosed with ALS or throat cancer prior to death. Like you said however, "rarity" is relative. What is rare today may change next year, especially if the disease is brought upon by some environmental condition. In addition, what is "rare" here at home may not be so rare elsewhere, so it largely depends on the region. For example, leukemia is rare in North America, but common in the Middle East, where depleted uranium from countless wars have lead to spikes in the number of cases

Diseases that necessarily cause death aside. How does one measure rare? Whenever I google search prevalence of particular diseases it might say it is uncommon or rare if there are fewer than 200,000 cases a year. I don't know if that number is completely arbitrary or what.