Both Cunliffe and the LSJ include both εἶδον (aorist, “saw”) and οἶδα (perfect, “know”) under the same rubric, which can be very misleading to intermediate students of Greek: yes, etymologically the Greek verb for “to know” at some point meant “to be in an enduring state that results from having seen,” but as a matter of actual usage it’s best to treat them as separate verbs.
1. The present forms never appear. The fake present form *εἴδω which Cunliffe and the LSJ use as a citation form is posited in order to explain a first aorist ἐεισάμην (on which see below)
2. The second aorist εἶδον always means “see.” (Or sometimes, in the middle, “seem.”) Its unaugmented stem is ἰδ-. Both in Attic and in Homer, the other tenses of “to see” are supplied by ὁράω.
3. The first aorist ἐεισάμην is only used in the middle (i.e., it is a middle deponent), and means “seem” — compare Latin videor.
4. The perfect οἶδα always means “know,” and is used as a present; there are likewise future perfect forms that are used in a future sense, and pluperfect forms used in a imperfect sense. Sometimes dictionaries will shortcut the confusing morphology by simply calling these forms “future” and “imperfect.” Its morphology is very irregular; the most common stem is εἰδ-, but ἰδ- and οἰδ- also occur. (Under the influence of certain consonants, δ may become σ in any of these stems.)
If it’s helpful for you to think of this as οἶδα lending its aorist to ὁράω, by all means do so.
A note on the most glaring ambiguity:
εἴδομεν, εἴδετε are either the indicative of εἶδον, or (in Homer) the short-vowel subjunctive of οἶδα. The subjunctive of εἶδον is ἰδωμεν, ἰδητε: long-vowel (because the second aorist is thematic) and derived from the unaugmented stem.