While analyzing this particular verse, the word used for moderation in the Koine Greek is "epieikies" meaning equitable, fair, and/or fitting. How does that translate to moderation or in some other translations gentle?
The operative adjective in Phil 4:5 is ἐπιεικής which occurs just five times in the NT - Phil 4:5, 1 Tim 3:3, Titus 3:2, James 3:17, 1 Peter 2:18. The cognate noun occurs in Acts 24:4, 2 Cor 10:1.
According to BDAG, this word means:
not insisting on every right of letter of law or custom, yielding, gentle, kind, courteous, tolerant
According to Phil 4:5, this characteristic of the Christian must be a conspicuous one; that is, Christians must be well-known for their gentle, kind, courteous and tolerant ways with others. Put another way, behave as Jesus would because this was one thing Jesus was known for.
Ellicott describes it this way:
(5) Your moderation.—The word here rendered “moderation,” properly denotes a sense of what is seemly, or equitable, as distinct from what is required by strict duty or formal law. Such distinction the world recognises when it speaks of what is enjoined, not so much by duty as by “good taste, or “right feeling,” or (with some peculiarity of application) by “chivalrous” feeling, or the “spirit of a gentleman.” Here it denotes the general sense of what is seemly in a Christian tone of character. In 2Corinthians 10:1 (where it is translated “gentleness”) it is ascribed emphatically to our Lord Himself. But the usage of the New Testament appropriates it especially to the “sweet reasonableness” which “gentleness” may well designate. Thus, in Acts 24:4 it clearly signifies patience, or forbearance; in 2Corinthians 10:1 it is associated with meekness; in 1Timothy 3:3, Titus 3:2, with peaceableness; in 1Peter 2:8, with kindness; in James 3:17 the word “gentle” is placed between “peaceable” and “easy to be entreated” (or rather, persuaded). This spirit is, no doubt, “moderation;” but it is something more. It may refer here both to the exhortation to unity in Philippians 4:1-3, and to the exhortation to joy immediately preceding. It would help the one and chasten the other.
Barnes suggests this:
Let your moderation be known unto all men - That is, let it be such that others may see it. This does not mean that they were to make an ostentatious display of it, but that it should be such a characteristic of their lives that it would be constantly visible to others. The word "moderation" - ἐπιεικὲς epieikes - refers to restraint on the passions, general soberness of living, being free from all excesses. The word properly means that which is fit or suitable, and then propriety, gentleness, mildness - They were to indulge in no excess of passion, or dress, or eating, or drinking. They were to govern their appetites, restrain their temper, and to be examples of what was proper for people in view of the expectation that the Lord would soon appear.
Definition: seemly, equitable, yielding
Usage: gentle, mild, forbearing, fair, reasonable, moderate.
Note that Strong distinguishes between definition and usage. Definition is the lexical/dictionary meaning. Usage is the pragmatic range of meanings found in actual sentences.
Where does the sense of "gentle" and "moderate" come from?
1933 epieikḗs(an adjective, derived from 1909 /epí, "on, fitting" and eikos, "equitable, fair"; also see the noun-form, 1932 /epieíkeia, "equity-justice") – properly, equitable; "gentle" in the sense of truly fair by relaxing overly strict standards in order to keep the "spirit of the law."
1933 /epieikḗs ("justice beyond ordinary justice") builds on the real intent (purpose) of what is really at stake (note the epi, "upon") – and hence, is true equity that appropriately fulfills the spirit (not just the letter) of the law.
In https://biblehub.com/philippians/4-5.htm, only King James Bible translates it as "moderation":
Let your moderation be known unto all men.
12 versions translate it as "gentle" or "gentlemess": New International Version
Let your gentleness be evident to all.
4 versions use "reasonable": English Standard Version
Let your reasonableness be known to everyone.
In anycase, Paul's idea is that don't be so strict and extreme about the letter of the law; be fair minded, reasonable, and gentle. This is a good lesson for us all today.