> Who cares if they're actually standing? They can lay on the floor if you ask me, as long as they can share the status with team members.

Standing up was a specific solution to a specific problem. There may be other ways to achieve the same thing, but you should be aware of what you're doing.

> Your standup should be about tasks, not about people. The difference is subtle, because it the end everyone should share the same info - progress, blockers, etc. But by checking on tasks the team members automatically build mental model of current iteration. How far are they? Is the tempo OK? Or maybe there's some delay and, finally, can they deliver on time?

This really doesn't seem such an important distinction. You assign tasks to people in your iteration planning; if a task is stuck it will be because a person is stuck on it and vice versa.