The original approach to lattices, following Boole, was algebraic rather than graphic or order-theoretic, and simple examples that quickly come to mind today were not really in the offering. The pioneers, Boole, Peirce and Schröder, operated with a fluid set of algebraic rules and logic related principles, and examples of logical systems they operated with were hard to visualize and distinguish from each other. Blyth likely refers to the episode with Peirce's casual remark in On the algebra of logic (1880) that was disputed by Schröder. It was instrumental in sorting out which sets of rules lead to which structures, and coining the lattice concept in particular. The relations between the syllogistic, propositional logic and "the logic of relatives" (predicate calculus that Peirce discovered independently of Frege) were clarified around the same time.
Ma and Pietarinen describe the episode as follows in Peirce’s Sequent Proofs of Distributivity:
"In the 1880 paper, Peirce stated the following distributive laws... He casually mentions that “they are easily proved . . ., but the proof is too tedious
to give” [11, p. 33]. This passage provoked a challenge by Schröder, who took the
laws of distributivity to be independent from the lattice axioms. A rejoinder and
a lively discussion ensued and Peirce’s lost and subsequently recovered version
of the proof was finally added to Huntington’s 1904 paper... Huntington published in 1904 the proof Peirce had sent to him,
including Peirce’s footnote about it. In the published proof, the axiom — Huntington’s “postulate” — number 9 is crucial. Without it, the axioms 1–8 only
define uniquely complemented lattices which need not be distributive.
Here is Peirce's own description from the follow-up paper On the algebra of logic (1885), where he admits slipping in a "non-syllogistic" principle (dilemmatic argument) into his proof of the "wrong" direction of distributivity ($-\!\!\!<$ is Peirce's symbol for the material implication):
"The dilemma was only introduced into logic from rhetoric
by the humanists of the renaissance; and at that time logic was studies with so
little accuracy that the peculiar nature of this mode of reasoning escaped notice.
I was thus led to suppose that the whole non-relative logic was derivable from the
principles of the ancient syllogistic, and this error is involved in Chapter II of my paper in the third volume of this Journal [the 1880 paper]. My friend, Professor
Schröder, detected the mistake and showed that the distributive formulae
$$(x + y)z\ -\!\!\!<\ xz + yz$$
$$(x + z)(y + z)\ -\!\!\!<\ xy + z$$
could not be deduced from syllogistic principles. I had myself independently discovered
and virtually stated the same thing. (Studies in Logic, p. 189.)"
Schröder developed what we would now call theory of lattices in Vorlesungen Über die Algebra der Logik (1891), the first systematic exposition of modern logic that laid the foundation of Russell's Principia. There he gave the first example of a non-distributive lattice, but it was not an elementary one one would think of today. Here is from Peckhaus's Mathematical Origins of 19th Century Algebra of Logic:
"In the sections dealing with statements
without negation he proves one direction of the distributivity law for
logical addition and logical multiplication, but shows that the other side can not be proved, he rather shows its independence by formulating a model in
which it does not hold, the so-called “logical calculus with groups, e. g. functional
equations, algorithms or calculi”. He thereby found the first example
of a non-distributive lattice."
