I came across this word, and couldn't find it in any dictionary.
Logicious does not appear in the 1971 edition of The Compact Edition of the Oxford English Dictionary, which, together with the continuing rarity of the word, constitutes strong evidence that it doesn't appear in any dictionary of well-established English words.
For fuller context, I offer the following larger excerpts from the OP's three cited sources of logicious. From the "Advertisement" to Thomas Branagan, A Preliminary Essay, on the Oppression of the Exiled Sons of Africa (1804):
With great deference does the author of the following pages submit them to the inspection of a discerning public. To novelty of sentiment, or refinement of composition, he does not pretend. From a slave-trader or a West-India-planter, neither the one nor the other will be expected. If his labour of love obtain the approbation of the genuine friends of religion and humanity, he is willing to allow logicious cavillers and snarling critics to say what they please. If it contribute, in any degree, to the promotion of the good cause in which he has embarked, his end is gained ; he is amply rewarded.
From Canadian Bookman, volumes 1–2 (1919) [combined snippets]:
Perhaps there was a special reason for his unattached condition:
CONSTANCY—I have heard him aver, is an authorized impertinence.
Was this bitterness, or only a playful perversion of logic! as thus:—"An authorized constancy is not impertinent. An unauthorized constancy must be impertinent." Was this a divided heart, or only, as the logicious say, an undistributed middle?
And from Peter Dent, Handmade Equations: Poems 2000–2004 (2005) [combined snippets]:
What then? the wild and the unspecific the truly
Unspeakable a perfect eradication but fill in the
Blanks get events no matter how ghostly to mat-
Erialise find that somewhere and someone 'neither
Logicious nor musicious' as a wicked wit had put it
Published in 1804, 1919, and 2005, each of the three examples is roughly a century apart from the next nearest one (or ones). With regard to the most recent instance, from a poem by Peter Dent, I agree with FumbleFingers's comment above that the use of logicious there is more a matter of word play (vaguely alluding to logic, just as musicious presumably alludes to music) than anything else.
The other two examples, however, seem to me to involve serious attempts to produce an adjectival form of logician. Neither of the two earlier instances of the word occurs in the context of any evident admiration for logicians or their arguments, so it may be that the authors were drawn to the form logicious because it carried a hint of the sound of such unadmiring words as capricious, officious, and meretricious. But whether that is the case or not, the use of a word meaning "characteristic of or pertaining to a logician" is entirely suitable in both of these excerpts.
Consider, for example,
...he is willing to allow caviling logicians, practiced in their art, and snarling critics to say what they please.
Was this a divided heart, or only—as logicians put it in their analytical patois—an undistributed middle?
In each case the meaning is much the same as in the wording that the author actually chose, but it lacks the flourish that the neologism provides.
I think that Branagan and the author of the Canadian Bookman excerpt use logicious to mean "characteristic of or pertaining to a logician." The word never caught on, but its meaning in these passages seems clear enough.