We laypeople use general terms where it starts: breast cancer, lung cancer, skin cancer. Metastatic when it invades other places, and we give it grades. Stage II, stage III, stage IV (and sometimes finer grades). General terms help us define what to focus on. This probably fits with your third point. For specific destinations, different cancers like to metastasize to specific places, but not always. We don't know why, it is still being investigated, but there are patterns.
Your second point is more advanced. Cancer cells have specific characteristics, in general we call them expressions or mutations, usually identified through biopsy and/or liquid biopsies. PD-L1, HER2, ALK+, PTEN, BRCA, BRAF, KRAS, PI3K, etc. We learn new ones all the time and sadly cancer will mutate and turn on/off characteristics. Cells are adaptable and duplicate with these adaptations.
I'm currently reading this book to help me understand (latest edition): https://www.google.com/books/edition/Molecular_Biology_of_Ca...