'Seamless' loop hangs/pauses at beginning of clip
Within an editor each frame is complete in itself, but in the output formats you mention, the video is almost always compressed in such a way that only certain frames (called I-frames) are complete and independent. The rest are 'difference' frames that rely on information from surrounding frames. 

At a loop point there's often a need for the player to start over, to refill its buffers and prime the engine. You can often help out by ending your segment with a I-frame. But the only sure way to replicate what you see in the editor is to output only I-frames. MJPEG is an example of a codec that always does this, but there are other codecs that can be instructed to. The downside is that the video will be considerably larger. 

Try MJPEG and see if it does what you need.