So right after I posted the above, I decided to try for the third time - except I first went to
https://java.com/en/download/ and updated Java on my computer (which I don't believe should have affected anything) - to version 8 release 91. Then I went back to the indexing download page through Firefox instead of Chrome (again, not expecting it to have any different result) - but it worked. I am now able to index.