In 1960, I introduced one of the first courses in what is now called Critical Thinking.
It was then called Critical Analysis resulting in a text and an international journal by the same name.
Since then, "Critical Thinking" has become a shibboleth of our educational institutions both on the college and pre-college levels and Critical Thinking texts have sprouted like weeds.
Unfortunately, there is not unanimity in the meanings of the term and there does not, yet, seem to be a delineation distinguishing clear, critical, and analytical thinking one from the other. I shall try to bring some clarity to the issue. However, after doing so, I will cede to society's penchant for using words loosely and meld the three terms into the most popularly used one, CRITICAL THINKING.
It should be understood, however, that critical thinking does not NECESSARILY or very often require analytical thinking.
Fundamentally, critical thinking is a complex of attitudes varying in intensity and use. If I may give a non-exhaustive list of some of the ways the mind manifests these attitudes, a critical thinker, at one time or another, but not always, is curious, questioning, speculative, creative, inquisitive, reflective, thoughtful, open-minded to evidence (i.e., to verifiable or falsifiable claims), perceptive, persistent, observing, resistant to gullibility and accepting absolutes,    interested in objectivity and rational discussion, prone to unambiguous definitions, sometimes analytical, and much more.
Not all of these attitudes function simultaneously. This accounts for the many different concepts of critical thinking. Teachers of different academic orientation, define critical thinking in terms of their own areas of expertise. They are not concerned with examining the fundamental assumptions upon which their interests rest--to mention a few:
Moralists and theists assume the real existence of moral principles and then apply critical attitudes to them.
Theists declare the existence of their unverifiable gods, and then proceed critically to describe them in detail.
Scientists are not concerned with examining their assumption that there is a physical world beyond sense data nor whether the language of science, i.e., mathematics, in fact describes reality.
Poets and artists, since they are not concerned with truth and knowledge as defined by scientists, think critically about choosing words or images that will stimulate personal meanings in those who view their works.
Mathematicians are not concerned with discussing the reality of their geometric concepts. As Bertrand Russell so aptly put it, "Pure mathematics is the subject in which we don't know what we are talking about nor whether what we're talking about is true." Often the critical thinking of a student of mathematics constitutes a clear and critical understanding only of the manipulation of little-understood concepts.
In ordinary language, a child who is told to "hand me that red pencil," and who clearly and discriminatingly understands which pencil to select, is not concerned with whether or not the pencil is in fact the color her perception "tells" her it is. It is evident then, that clear thinking may require only a common acceptance of unexamined assumptions as is so often the case in social discourse.
In other words, too many critical thinkers are either not aware of their assumptions or are not interested in examining them or the complexities of language, truth, and knowledge as an analytical philosopher would be.
Now to ANALYTICAL THINKING which necessarily involves critical thinking. But there are, of course, degrees of analysis also. Generally, it is analytical thinking that requires not only an intense critical attitude but also a body of facts and a method to which the critical attitudes are applied.
The ultimate in analytical thinking requires an extensive study of facts, laws, principles, rules, and more particularly the use of unequivocal nit-picking and specificity of definitions.
When a mathematician declares that there are as many points on a hypotenuse of a right triangle as there are on its legs, an analytical thinker with a critical attitude will be interested in the definition of a point in order to have greater insight about mathematical language and its implications. When it is cited that a point has no dimensions (one of its various definitions), his analysis of the language indicates that, in a circuitous way, it is in fact saying that there are as many nothings (IDEAS of a point) on the hypotenuse as there are on a leg. This should give him a clearer idea of the nature of math.
When a theist declares that God is all knowing, an analytical thinker, having had God defined to be incorporeal, examines the theistic language and critically asks, "How can an entity that is not matter and consequently has no brain, be capable of knowing anything?"
To be an analytical thinker, one must learn many facts about the world and particularly many fundamental principles such as:
Learning to recognize assumptions and when a discussion requires that they be examined.
No word or symbol has an inherent meaning.
All meaning occurs ONLY within functioning brains.
No two sentient beings experience the world in the same way.
Knowledge is experience but experience is most often not knowledge.
All systems of language including logic, mathematics, geometry, and even ordinary language are systems of unspoken IF---THENS.
According to available evidence, a fundamental characteristic of knowledge is that it is only probable.
Truth is a function of language; if there is no language, there can be no truth or falsity.
Claims that are "true by definition" are unverifiable and unfalsifiable and, hence, do not relate to the physical world that we sensibly presume exists beyond our perceptions.
Majority opinion and longevity of an idea are not evidence of truth. Truth cannot be determined by a vote.
And finally, one must not take the terms 'language', 'truth', and 'knowledge' at face value.
These terms name three of the most complex concepts that human beings have to deal with every day of their lives. And it is the abuse of those terms all over the world that contributes so greatly to its ills.
The above are only a few of the attitudes, important facts, and principles involved in being a CRITICAL THINKER in the social, i.e., conventional, definition of that term.