Is there a way to save a multi-page indd document set up in printer spreads so it doesn't carry the bleed from the next page onto the pdf?

Setting the inside bleed to 0 is the correct answer, however if you want inside bleed for a wire bound book and you want to fix a specific page, you can turn off "allow pages to shuffle" in the pages panel fly-out menu, then drag the right side spread page away from the left until it drops down to a single page, then extend the bleed on those pages.

In most cases, you can fix this issue by setting your inside bleeds to zero.

You don't need to bleed toward the spine unless you are printing as single pages or for wire binding.

Please elaborate with graphical examples!

I suppose this is more an Indesign question than a Acrobat printing question.

The bleed in Indesign is created according to the design of your Indesign file. If there is bleed data, it is there, because you asked to output it like that.