First one looks a lot like Chinzan. The one in your image seems more compact that Osakazuki would be. It definitely is a satsuki very close to Chinzan or Osakazuki.
I would measure the flower size and if it matches, I would just call it Chinzan (plus it was sold to you as such). Because Chinzan/Osakazuki are a bit the generic pink species form or R.indicum, you have some leeway, I feel. Especially with Osakazuki, it is said that in Japan there are two distinct forms that are very similar that have the same name.
The second one does not seem Kazan. Notice the stamen count. What is the average if you count many flowers? This looks very much like an R.nahakarae or North Tisbury variety.
R.indicum has 5 stamen. R.nakaharae has 10. R.nakaharae hybrids have a range from say 6 to 10, depending on the flower.
They should bloom later than satsuki. And they are creeping if you just let them grow like they wish.
The unique thing about Kazan is the very triangular-shaped leaves (there is two Kazan, but we are both talking about the dwarf red one). And it should grow even more compact than Chinzan. Sadly, I never saw a Kazan in person. It must be really nice. The downside is the slow growth rate. But you can't have one (compact growth) without the other (slow growth).
Kazan's very unique foliage:
Kazan Azalea is a slow-growing broadleaf evergreen shrub with green foliage. In spring pink flowers emerge. Features glossy texture. Attracts bees and butterflies making it an excellent addition to pollinator gardens. Grows well with mostly sun - mostly shade and even moisture - regular water...
plantlust.com
If I had a rock garden, Kazan would be the first plant I would put in it.
