When tiling a floor must I start in the middle of the room?
The two things (starting in the middle and avoiding slivers) are NOT related.

You can start in the middle and end up with idiotic cuts on both edges - or you can start in the corner and not do that...

What you SHOULD do is measure enough that you know where to put the SENSIBLE cuts on the edges, regardless if you are starting at a wall, corner, or the middle of the floor. When starting in a corner or at a wall, your very first tile might well be a cut tile. By planning, you know what cuts you are making and can place the whole and cut tiles as you like. In your case, the first two-and-a bit tiles at the door should probably be cut so that there's not a large gap at the threshold (or you need to get the door molding cut so they can slide under it.) Depending how the (invisible) far wall (opposite the door) comes out, you might want to take enough off the "door" tiles that the wall parallel to the door does not have what looks to be a 1-1/2" slice running along it.

The main argument I've seen for starting in the middle is to avoid following an irregular wall - but you can do the same thing by striking a chalk-line where the first grout line from the wall would be, and pre-cutting all the tiles for that wall if it's irregular (or if they will be half or 1/3 tiles so the far wall isn't slivers - which you'll know because you measure it.)

Based on your pictures, I'd choose the "not centered" approach, since the visible far wall is decidedly not slivers that way, but that's purely an opinion call, not "right" or "wrong."