I just want to thank Andrew most sincerely for his reply. I had hit the books over this problem, and eventually after many hours of frustration eventually tried the solutions he's just suggested, but still couldn't print the Indesign file as shown on the screen or match how the graphic showed when printed from Illustrator (but did learn alot in the process I must admit). However something I read on this site made me look into the opacity settings on my Illustrator graphic, so as a test I changed them to being darker, then sent it to Indesign and printed it, and low and behold, Indesign liked those better and printed the graphic darker where I'd wanted it to be. I don't know why Indesign handles the opacity percentages received from Illustrator differently when printed, but I'll remember in future that it does. If only I'd have asked my original question sooner, I would have saved a lot of frustration and time, and I'm pretty sure I'd have tumbled to the opacity solution much earlier. Ah well, got there in the end.
Thanks again Andrew.