It is much harder than it might seem to answer this question and find recommendable "books that introduce to a layman science and its method/spirit".
The subject is immense: it covers all theories and all observations: making it difficult or impossible for books on scientific method to digest entirely and present their subject. Some books concentrate on theory, others, perhaps a valuable minority, on the way theories are built from the ground upwards on observation and experiment. Some achieve a degree of brevity but pay a heavy price in accuracy.
In view of these difficulties, an author who can extract an important gist briefly from them has an enviable skill, and so I would recommend first, in spite of its age, a short essay by :
Fernando Sanford (1899) -- 'The Scientific Method and Its Limitations' ~ 20pp. (Stanford University Press) [https://stacks.stanford.edu/file/druid:mp840zw9344/SC1020_1899.pdf]
Sanford's brief essay dated the modern awakening of scientific investigation in Europe to about the year 1600, and especially to the works of William Gilbert on magnetism, and of Galileo on moving (falling) bodies. Sanford explained why he considered their innovations in method to be so fundamental, and described important features (and difficulties) of carrying out experimental science more generally. He did refer also to some (now) very dated and dead controversies, and the world has moved on unimaginably since 1899, but still much of what Sanford wrote about scientific principle and method remains of value. Much more could of course be and have been said, but in the meantime the essay can still deliver briefly a core of material to "introduce to a layman science and its method/spirit".
One of the almost perpetual puzzles about science is the question, what degree of confidence, or what certainty if any, can we place in a theory and belief, and what circumstances should we take to reinforce or undermine that confidence? The spirit of science is one of critical assessment, and at its foundations ideas of probability. The history of what the idea of 'probability' has meant has shown radical change. Thus I would also recommend:
Ian Hacking's 'The Emergence of Probability: A Philosophical Study of Early Ideas about Probability, Induction and Statistical Inference' (1975), [https://books.google.com/books?id=Z2g0F1V5DywC],
a book that insightfully shows important historical antecedents and pre-modern underpinnings of current attitudes towards indicative signs, facts, and evidence.
Isaac Newton's work for a long time had considerable influence in forming ideas of what scientific method is and should be. In spite of all that has happened since, there is much in what Newton did, and in what his successors made of it, that still illuminates "science and its method/spirit". The questioner's "laymen" will find much of that method and spirit in I Bernard Cohen's historical study of the experimental side of Newton's legacy and its connection with that great experimentalist Benjamin Franklin:-
I Bernard Cohen, "Franklin and Newton", Memoirs of the American Philosophical Society vol.43 (1956), repr. 1966;
and also
"The Cambridge Companion to Newton" (2002), eds. I.Bernard Cohen, George E. Smith.
Equally and still relevant to the method and spirit of science are questions posed and answered in:
(a) Steffen Ducheyne (2012) "The Main Business of Natural Philosophy: Isaac Newton's Natural-Philosophical Methodology";
(b) William L. Harper (2011) "Isaac Newton's Scientific Method: Turning Data into Evidence about Gravity and Cosmology"; and
(b) George E Smith's "Closing the Loop: Testing Newtonian Gravity, then and now", which has appeared in several forms (essay and lecture transcript), most recently at pp.262-352 of "Newton and Empiricism" (2014), eds. Zvi Biener & Eric Schliesser.
To close, a word about one of the books given in the questioner's list:--
Barry Gower's (1996) "Scientific Method -- A Historical and Philosophical Introduction"
is a broad historical survey by a professor in the philosophy of science. It stretches in coverage from Galileo, Bacon and Newton to Reichenbach, Popper and Carnap.
There are many riches in this book, but it is worth mentioning that it shares one characteristic of numerous modern philosophical approaches to theories of science, a strong emphasis on theoretical problems such as skepticism and the status of induction. The result can sometimes be a great distance between the theory of philosophy of science and the practice of science and of scientific theorists.
For example, I could not imagine that any metallurgist for thousands of years has ever had reason to doubt that copper can be melted: but the chosen remit of this author's book makes it relevant for him to consider at some length, for example, Reichenbach's assertion that "because the conclusion that all copper is fusible is inductive and we have yet to establish whether such conclusions are acceptable, we cannot claim that it is true that all copper is fusible, or even that it is probable that all copper is fusible."
Clearly it is up to each prospective reader to decide whether this distance from the real world of scientific theory and practice brings the book's subject-matter closer to or farther away from the field of his/her interest.
