|An 1858 description of an "organized body", in context of the main 39 points of general chemistry, from English chemist John Bidlake's Textbook of Elementary Chemistry, which differentiates all element-based "bodies" into two types: organic (bodies which live) and inorganic (bodies which don't live); which is a defunct precursor model, to say, to the modern 21st century ecological stoichiometry view of the differentiation of all element-based "bodies" into two main types: animate (bodies which possess the property of animation) and inanimate (bodies which do not possess the property of animation); hence, the name distinctions of animate matter vs inanimate matter, animate organism vs. inanimate organism, animate chemical vs inanimate chemical, etc. |
The following 1860s letter by Henry Adams, with its mention of "animated being", might well capture the meaning of the term animate organism: 
“Everything in this universe has its regular waves and tides. Electricity, sound, the wind, and I believe every part of organic nature will be brought someday within this law. The laws which govern animated beings will be ultimately found to be at bottom the same with those which rule inanimatenature, and as I entertain a profound conviction of the littleness of our kind, and of the curious enormity of creation, I am quite ready to receive with pleasure any basis for a systematic conception of it all. I look for regular tides in the affairs of man, and, of course, in our own affairs. In ever progression, somehow or other, the nations move by the same process which has never been explained but is evident in the oceans and the air. On this theory I should expect at about this time, a turn which would carry us backward.”
The 1620s reaction automaton theory discussions of French philosopher Rene Descartes (IQ=195), the 1900 “I am an automaton endowed with power of movement, which merely responds to external stimuli beating upon my sense organs” theory of Nikola Tesla (IQ=195), along with the 1950s free energy electochemical automaton theory of John Neumann (IQ=185) may well facilitate the differentiation of the various types of "animation", e.g. Greek automaton animation, AI-based animation, etc., as compared to the animation found in a moving reactive human.
Greek philosopher Aristotle (IQ=195) freely used the term "organon", spelled "organ" in English, to denote a part of a living being or of man in particular. 
The term “animate organism”, in the modern hmolscience perspective, is an upgrade from the defunct term “living organism”, in that a body can possess animation or reactivity, but not life—in the sense that “life” is something that does not exist, in the famous 1925 words of Nikola Tesla.
To explain another way, in the hmolscience periodic table perspective, a zirconium-based organism, e.g., will not have the property of "animation"; such chemical species may exist, and may have the property of trajectory or unidirectional movement, possibly even spin, but not complex animation. Conversely, carbon-based organisms, as embodied in the out-dated term "carbon-based life", will tend to possess the property of animation; but not all carbon-based entities, graphite being one of many examples. Those carbon based animate organisms in possession of the so-called CHNOPS-based or "CHNOPS system" set of core elements, while in the bound state of animate existence, may well then be classified as types of "animate organisms" and as such function to proactively take the place of the now-defunct terms of olden: living organism or living system, which have no hard physical science basis, but only mythological basis; hence resulting to do away with the "unbridgeable gap" model of the last century.
The simple 3-element retinal molecule is a simple example of a animate organism, one moved by the exchange force of the photon; the 15-element bacteria molecule is an intermediate in complexity example of an animate organism, one that generally requires some type of agar medium to function, DTA, driven in surface-attached motion by a heat, is another example; the 26-element human molecule (person) is a more complex example of an animate organism. Other examples include: fish molecule, cell-as-molecule, the two-legged kinesin protein molecule, that walks along microtubules while carrying cargo.
Unbridgeable gap models
The above defunct theory of life-based "animate organism model" is the modern and correct view. Some, however, even in modern times, may wish to retain some semblance of autonomy, free will in the form of self-organization, or ontic opening stylized model of purpose, and will attempt to take recourse in the "unbridgeable gap" model and will attempt to explain the "start of life" using some type of chemical perpetual motion conception, often using feedback loop models (e.g. autocatalytic closure). These models, however, tend to yield absurdities, one after another, paragraph after paragraph, in a spiral down a path towards inanity. One example of this, is the 2007 book The Self-Creating Universe, by American chemistry-trained dermatologist David Alkek, who attempts to employ the term "animate organism" in the context of chemistry, in his incorrigibly-entitled chapter "self-reproducing matter", as follows:
"Biology can never be reduced to chemistry and physics. There is a definite step between animate and inanimate. What is an even greater mystery is the progression from inanimate mass of chemical to an animate organism, capable of independent existence and replication. We don't have to involve a transcendental power to do this. The emergence of living from nonliving matter is another example of the self-organizing power of matter and energy."
What Alkek has done, in short, is transferred "god theory", i.e. transcendental power", into the terms "emergence" and "self-organization", nothing but waterbasket terms.
The reader is more than welcome to tread through the inanity of the pages preceding this wondrous passage, inclusive of citing Robert Hazen’s 2005 book Genesis as citing American RNA-based life theorist Gerald Joyce’s famous 1991 working-definition of life as: “a self-sustained chemical system capable of undergoing Darwinian evolution”; mention of the Urey-Miller experiment (1952); mention of the meteorite theory of life; mentioning that “viruses blur the border between living and nonliving matter”, and so on down the yellow brick road.
German mathematical philosopher Edmund Husserl (1859-1938), according to Google Books search results, seems to have been a dominate advocate of the term “animate organism”.  Danish natural science philosopher Claus Emmeche uses the term “animate organisms” within the loose context of thermodynamics discussions in his 2004 article “A-life, Organism and Body: the Semiotics of Emergent Levels”; inclusive of all the other “life terms”. 
● Animate engine
● Animate system
● Animate thermodynamics
● Chnopsology | the study of CHNOPS-based animated structures.
1. Husserl, Edmund, Welton, Donn. (1999). The Essential Husserl: Basic Writings in Transcendental Phenomenology (animate organism, 13+ pgs). Indiana University Press.
2. Emmeche, Claus. (2004). “A-life, Organism and Body: the Semiotics of Emergent Levels”, in: Mark Bedeau, Phil Husbands, Tim Hutton, Sanjev Kumar and Hideaki Suzuki (eds.): Workshop
and Tutorial Proceedings. Ninth International Conference on the Simulation and Synthesis of Living Systems (Alife IX) (pgs. 117-24), Boston Massachusetts, September 12th, 2004.
3. Bidlake, John P. (1858). Bidlake’s Elementary Chemistry: a Text-book of Elementary Chemistry for the Use of Schools and Junior Students (§2: Chemical Affinity, pgs. 15-23, synthesis, pg. 6; number of elements, pg. 8). London: Allman and Son.
4. History of the term ‘organism’ – English.StackExchange.com.
5. Alkek, David S. (2007). The Self-Creating Universe: A Synthesis of Science, Philosophy, and Religion Creating a Theory of Universal Existence ("animate organism", chemistry, pg. 34). iUniverse. (ebook). LuLu.com.
6. (a) Taylor, Matthew A. (2008). Universes Without Selves: Cosmologies of the Non-Human in American Literature (pg. 108), PhD dissertation, Johns Hopkins University. ProQuest, 2009.
(b) Adams, Henry. (date). “Letter”, The Letters of Henry Adams (I:395-96), ed. J.C. Levenson, et al. Harvard: Cambridge University Press.