Broken back to English, it seems to go like this for ideals in commutative rings:
1 Prime ideals are of the form nZ, where n is a prime.
2 Primary ideals are composed of powers only of a prime element. This means n, the prime element, is also primary, even with the lowly power of 1.
3 Semiprime ideals are combinations of more than one prime, but which are also square-free.
I heartily agree with Wildberger that more good examples are needed. The Wiki-pega article on Semiprimes is valuable just because it gives a specific example, which cuts off a lot of questions at the pass. The article notes with the required specificity for dummies that 30Z would be a semiprime ideal, where as 12Z would not be. Mathematicians act like specific examples are going to kill them or lower their princely standards. The example makes it clear that 30 is semiprime because its factors are no more than single powers of primes.
For myself, adjusting to the language of ideals will take further familiaization to become entirely comfortable. Experts often talk somewhat loosely among themselvs, and tend to continue this trend in their expositions. Most mathematicians are poor expositors when it comes to bringing their abstract notions out of the darkness for laymen or even interested amateurs.
There is a reason for this: the second job is more formidable. With another expert, talking over concepts is easy. As Wildberger notes somewhere, it is basically Santa Claus to the Easter Bunny power, a pure manipulation of symbols. When I am done, I will be able to make the notion of ideals and their ramifications clear to an interested person. If it is clear to me, I should be able to do that.
By the way, I am looking at 12Z. It is not a prime, it is not primary, and it is not semiprime either. It is white on the diagram. It must play the role of a strict composite in ideals. I don't know, I am just guessing. I will overcome many impasses and wrong notions as I continue to chip away. In the end I have to be able to perform the arithmetic of ideals as easily as I can perform modular arithmetic in integers.