That page you link to is correct, wood movement is almost completely across the grain. Longitudinal or long-grain movement is so slight it can nearly always be ignored.
Just to note, we must remember movement is both expansion and contraction, which typically happens seasonally:
Obviously there are specific instances where movement cannot be accurately predicted. Where the grain shifts direction, as much as 90° within the same board (e.g. wood taken from various crotches within the tree) and particularly where there is wildly changing grain (e.g. in a burr or burl).
Are there particular orientations which I should avoid, e.g. radial in one piece in a plane with tangential of another?
This is the standard cross-grain situation you want to try to avoid where possible. So the answer is basically yes but it's complex because you can't avoid it in many cases.
Where you have no choice but to make a cross-grain join, you must build in some means to allow for movement, for example by fixing loosely as with various tabletop fasteners, fixing at one end of the joint only or allowing something to float within a groove or rebate (US: rabbet), as in the floating panel in frame-and-panel construction.
Unless you take these sort of steps to avoid an issue you can take it that something will happen, not that it might.
With what orientations is it safe to use screws without hardware that would allow movement?
That depends for a start on how you define hardware. The standard 'buttons' used to attach tabletops and allow movement are wood, and are made by the woodworker, but are technically hardware.
Rather than think about the orientations it is safe to use screws visualise the expansion and contraction of the wood and how you allow this to happen after fixing. So essentially screws could be safe in any joint they can be used in, as long as they're utilised appropriately.
A good example is the top to any table, which typically will be cross-grain to some portion of the underframe or apron assembly. One standard way is to fix firmly in the centre and allow the front and back edges to move. Where you want an edge to remain fixed relative to the piece (e.g. a table intended to go flush against the wall) you fix firmly there and allow movement to occur out from that edge:
Screws are used here both for the points where firm fixing is desired as well as where movement is being allowed for, by screwing directly into the tabletop through the apron and through some form of movable fixing/fastener respectively.