A knot is tiable
in-the-bight of a line, means that we can tie ( and untie ) it, without using any of its ends - without tucking and/or un-tucking them throughout any opening of the knot, at any time. However, this does not mean, of course, that we can not tie this knot NEAR the end of the line !
And it also does not mean, as some people seem to believe, that its two ends go to / come from opposite directions = that is can only be tied somewhere in the middle of a straight, tensioned line. Where the two ends of a tiable in-the-bight knot are pointing to, is a matter of
geometry - TIB is a matter of
topology.
A knot is tiable
with-a-bight of a line, means that, on top of the knot being tiable in-the-bight, we can tie ( and untie) it using one ( or more ) bight(s), as double line(s) with an accessible end : the tip(s) of t bight(s). This is a restriction of the general TIB condition, and the knots that are tiable with-a-bight are far fewer than the knots that are tiable in-the-bight. All the former can be tied as the later, but not vice versa. So, when a knot is also tiable with-a-bight, we know more things about its topology, than when it is just tiable in-the-bight. We can say that it is one (or more) double-line stopper(s), tied on the middle of the same line.
Generally speaking, we prefer to have a knot that can be tied with-a-bight, because we just use the bight(s) as double line(s) with accessible end(s) - the tip(s) of the bight(s). If we can not do this, but we have to do other, more complex manipulations of the segments of the rope to tie it in-the-bight, the tying method/procedure seems more complex, and more difficult to be memorisable and remembered. The simplest loop that can be tied with-a-bight, is the
overhand loop, while the simplest knot that can be tied in-the-bight, but NOT with-a-bight, is the
slipped overhand knot ( often used as loop in the
Trucker s hitch ).
( At the attached picture, a tiable
with-the bight(s) double loop (
two bights ).