I'm sure you'll get some more responses, but I'll give you my basic summaries of these terms.
Natural history is often described as the study of organisms in their environment with the goal of understanding how the organisms' interactions with their natural surroundings influence their behavior, forms, function, and abundance. Natural history can help us to understand an organism's evolutionary history and ecological interactions.
Life history is the study of organism reproductive strategies and traits. In a sense, life history is a subset of natural history focused on understanding how different reproductive strategies might influence fecundity overtime which has ultimate consequences for evolution. Examples of life history traits include age of first reproduction, lifespan, and number vs size of offspring.
The life cycle of species is the full suite of stages and forms an organisms goes through over its lifespan. Life cycles can be an important influence on life history strategies but the study of life cycles does not have to be focused on understanding organism fecundity.
I aprecciate your contribution. In someone papers i've reviewed, i found is not there consensus when work insects as Lepidoptera or Hymenoptera. Because someone authors aparently confused this concepts, others often use as sinonyms.
A life cycle is a period involving all different generations of a species succeeding each other through means of reproduction, whether through asexual reproduction or sexual reproduction.
In addition to reproductive traits, the life history of a species can also include growth in length and weight with age, length-weight relationship, sexual dimorphism and more. This is my understanding of the term 'life history', and these traits are often included in life history papers (at least for life history of fishes).
This is a pet peeve of mine. I often see papers that describe traits as life history traits when in fact they are behavioral traits or other traits that could be described as traits describing the natural history of an organism. There is an entire field of study of life history traits that define these traits as those that directly affect the demography of a population as described by the life table. Thus, these are traits such as size at birth, the age and size at sexual maturity, offspring size and number, age-specific schedules of reproduction and life span. Together these traits determine fitness and so are the primary targets of selection. It is important to be precise in our language, otherwise the words themselves become meaningless.