As of [utility.arg.requirements] Table 17/18, the return types of the expressions

a == b

or

a < b

for types satisfying the EqualityComparable or LessThanComparable types, respectively, are required to be "convertible to bool" which corresponds to a copy-initialization context. But several newer parts of the library that refer to such contexts have lowered the requirements taking advantage of the new terminology of "contextually convertible to bool" instead, which corresponds to a direct-initialization context (In addition to "normal" direct-initialization constructions, operands of logical operations as well as if or switch conditions also belong to this special context).

One example for these new requirements are input iterators which satisfy EqualityComparable but also specify that the expression

a != b

shall be just "contextually convertible to bool". The same discrepancy exists for requirement set NullablePointer in regard to several equality-related expressions.

For random access iterators we have

a < b contextually convertible to bool

as well as for all derived comparison functions, so strictly speaking we could have a random access iterator that does not satisfy the LessThanComparable requirements, which looks like an artifact to me.

Even if we keep with the existing requirements based on LessThanComparable or EqualityComparable we still would have the problem that some current specifications are actually based on the assumption of implicit convertibility instead of "explicit convertibility", e.g. [unique.ptr.special] p3:

template <class T1, class D1, class T2, class D2>
bool operator!=(const unique_ptr<T1, D1>& x, const unique_ptr<T2, D2>& y);

-3- Returns: x.get() != y.get().

Similar examples exist in [unique.ptr.single.dtor] p2, [unique.ptr.single.asgn] p9, [unique.ptr.single.observers] p1+3+8, etc.

In all these places the expressions involving comparison functions (but not those of the conversion of a NullablePointer to bool!) assume to be "convertible to bool". I think this is a very natural assumption and all delegations of the comparison functions of some type X to some other API type Y in third-party code does so assuming that copy-initialization semantics will just work.

The actual reason for using the newer terminology can be rooted back to LWG 556. My hypotheses is that the resolution of that issue also needs a slight correction. Why so?

The reason for opening that issue were worries based on the previous "convertible to bool" wording. An expressions like "!pred(a, b)" might not be well-formed in those situations, because operator! might not be accessible or might have an unusual semantics (and similarly for other logical operations). This can indeed happen with unusual proxy return types, so the idea was that the evaluation of Predicate, BinaryPredicate ([algorithms.general] p8+9), and Compare ([alg.sorting] p2) should be defined based on contextual conversion to bool. Unfortunately this alone is not sufficient: In addition, I think, we also want the predicates to be (implicitly) convertible to bool! Without this wording, several conditions are plain wrong, e.g. [alg.find] p2, which talks about "pred(*i) != false" (find_if) and "pred(*i) == false" (find_if_not). These expressions are not within a boolean context!

While we could simply fix all these places by proper wording to be considered in a "contextual conversion to bool", I think that this is not the correct solution: Many third-party libraries already refer to the previous C++03 Predicate definition — it actually predates C++98 and is as old as the SGI specification. It seems to be a high price to pay to switch to direct initialization here instead of fixing a completely different specification problem.

A final observation is that we have another definition for a Predicate in [thread.req.paramname] p2:

If a parameter is Predicate, operator() applied to the actual template argument shall return a value that is convertible to bool.

The problem here is not that we have two different definitions of Predicate in the standard — this is confusing, but this fact alone is not a defect. The first (minor) problem is that this definition does not properly apply to function objects that are function pointers, because operator() is not defined in a strict sense. But the actually worse second problem is that this wording has the very same problem that has originally lead to LWG 556! We only need to look at [thread.condition.condvar] p15 to recognice this:

while (!pred())
  wait(lock);

The negation expression here looks very familiar to the example provided in LWG 556 and is sensitive to the same "unusual proxy" problem. Changing the [thread.req.paramname] wording to a corresponding "contextual conversion to bool" wouldn't work either, because existing specifications rely on "convertible to bool", e.g. [thread.condition.condvar] p32+33+42 or [thread.condition.condvarany] p25+26+32+33.

To summarize: I believe that LWG 556 was not completely resolved. A pessimistic interpretation is, that even with the current wording based on "contextually convertible to bool" the actual problem of that issue has not been fixed. What actually needs to be required here is some normative wording that basically expresses something along the lines of:

The semantics of any contextual conversion to bool shall be equivalent to the semantics of any implicit conversion to bool.

This is still not complete without having concepts, but it seems to be a better approximation. Another way of solving this issue would be to define a minimum requirements table with equivalent semantics. The proposed wording is a bit simpler but attempts to express the same thing.