The former encompasses situations like contract development, signal processing, control & embedded systems, compilers, maintenance programming, etc. I've found that strictly typed languages like Haskell, Ada, or Rust can be very helpful here, even for exploratory programming. In this situation, you know (and have little control over) the data that goes into your program and the requirements that it must meet, and if it doesn't do so, it's because you made a mistake somewhere.
But many types of exploratory programming involve unknown inputs or unknown outputs. Think of data mining or scientific computing, where the issue is often that you have unclean data and your task is to figure out how it's messed up and how to fix it as quickly as possible. Or UI software (both web and mobile) and tech entrepreneurship, where the big unknown is often how the user will react to the software and where they will find value in it. I've found that dynamic languages are great for this, because the whole point is you don't know what the data will look like, let alone how it will be structured.
It was interesting to me that the two alternatives you mentioned were "write lots of little tests" or "let the type system guide what needs to be fixed". For a lot of the exploratory programming I've been doing lately, the most efficient approach is just "let it break". Go make your changes, and if some part of the system doesn't work anymore, just take note of that. Because the act of letting things break gives you important information. If you don't care, then obviously that feature wasn't very important to the system to begin with, and you should rip it out entirely. If you do care, you should fix it, and then write some tests to make sure it doesn't break again. This way, you can build a rough rank ordering of how important features are, so that when you're faced with engineering trade-offs (and you always are), you have a solid, experience-based intuition about what you really need to support and what you can let slide.