Safe or Out?

[Andy]

I need a lot more help to visualize this and try to answer your question...

you say:

Throw comes to the 3B ahead of the runner

then later:

3B catches the ball "off the back" of the runner

Which is it? Did the 3B have the ball at third before the runner got there or did the throw from the outfield hit the runner in the back? This is critical information needed in order to answer your question.

Understanding: Defensive player entitled to defend the base and permitted to make attempt at fielding play. Runner should slide - safety...

Your understanding is also incorrect. The defensive player may block access to the base only if he has or is about to receive the ball (about to defined as the ball being closer to the fielder than the runner is). The runner is not required to slide, he is only required to attempt to avoid contact.

This play is definitely a HTBT (had to be there), but your description is contradictory and makes it difficult to comment on what the umpire on the field may have been thinking.