Rhythm, tempo, and meter are all different aspects of time. Meter and rhythm both describe the distribution of certain events over time, but meter is generally concerned with defining evenly spaced points in time as a framework for describing rhythm. Events described by rhythm do not need to be -- and often aren't -- evenly spaced.
Tempo describes the speed at which the events happen. That is, it determines the magnitude of the temporal spaces between the events.
In my experience "time" is a common synonym for meter, as in "six-eight time," or perhaps tempo, where for example "march time" implies not only a time signature of 2/4 but also a tempo of 120 quarter notes per minute.
The most common use of "rhythm" in my experience is to describe, for example, the difference between, for example, a pattern of three quarter notes as opposed to a pattern of a half note followed by two eighth notes or a dotted quarter followed by an eighth and a quarter. Each of these is a different rhythm, while they could have the same pitches and be used in the same meter at the same tempo.
Without knowing the titles of the books, it's difficult to say more about whether the authors' use of the word "time" is idiosyncratic or misleading.
The Rhythmic Structure of Music by Grosvenor Cooper and Leronard B. Meyer, The Philosophy of Rhythm by Peter Cheyne & New Musical Resources by Henry Cowell. None of them explicitly define all of the elements as aspects of Time except maybe The Rhythmic Structure of Music, because they do categorize them as Pulse, Metre, Rhythm, and Tempo
I don't know these books but I'll hazard a guess: pulse and meter are very similar, but these are both fairly low-level concepts. Pulse is just a series of evenly spaced moments in time with no properties other than tempo. Meter is a (usually periodic) organization of pulse (groups of two or three, for example) in terms of a specified hierarchy (that is, the pulses may be subdivided by a faster pulse that is 2 or 3 times more frequent). Again, tempo is primarily the speed at which things happen.
These are analogous to graph paper or to a grid that an artist might trace onto a canvas to help with perspective in a landscape. They are much more in the way of a framework onto which a composer can hang a rhythm. Rhythm is much more variable than meter, tempo, or pulse, and it is therefore the foremost parameter of time that a composer can use as an avenue of expression.
Most theory and composition textbooks speak of pitches: how do you put pitches together at the same time to create a harmony? How do you put pitches together one after another to create a melody? How do you put groups of pitches together one after the other to create a harmonic progression? But such programs of study frequently give little or no attention to the temporal aspects of these sequences of pitches or harmonies. They may mention "harmonic rhythm" -- the pace of chord changes, which can speed up or slow down during a phrase -- and they may point out instances where harmonic rhythm is used for some effect, but they rarely provide a coherent theory that one can use to analyze these considerations or to apply them to one's own compositions.
