APL (Dyalog Unicode), 14 bytes^SBCS

Anonymous tacit prefix function.

'''∘⍋⊃¨⊂'∘⍋⊃¨⊂

⊂ enclose argument (to act on it as a whole)

…⊃¨ from that, pick one character for each of the following indices:

∘⍋ the indices that would sort the argument in the the order given by the following string (all non-members go in order of appearance at the end):

'''∘⍋⊃¨⊂' the characters '∘⍋⊃¨⊂

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J, 14 bytes

Anonymous tacit prefix function.

]/:']/:''i'i:]

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] the argument

…i: last occurrence (non-members get the index beyond end of lookup string) of each character in:

']/:''i' the characters ]/:'i

…/: use that to sort:

] the argument

05AB1E, 24 22 21 bytes

Σ"Σ"'"«"'«Rrk}"«Rrk}R

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Explanation:

Σ } # Sort
 "Σ" # Σ string literal
 '" # " string literal
 « # Concatenate last two literals
 "'«Rrk}" # '«Rrk} another literal
 « # Concat again
 R # Reverse literal (so: }krR'«'"Σ)
 r # reverse stack 
 k # find the current sorting index in our code string 
 R # reverse our sorted string

First time trying stuff in 05AB1E so probably lots to be saved

Jelly, 14 bytes

“;fɓḟṾⱮ”ṾɓfⱮ;ḟ

A full program accepting a (Python formatted) string which prints the output.

(as a monadic link it yields a list of lists of characters)

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How?

“;fɓḟṾⱮ”ṾɓfⱮ;ḟ - Main Link: list of characters, S
“;fɓḟṾⱮ” - list of characters = [';', 'f', 'ɓ', 'ḟ', 'Ṿ', 'Ɱ']
 Ṿ - un-evaluate = ['“', ';', 'f', 'ɓ', 'ḟ', 'Ṿ', 'Ɱ', '”']
 ɓ - start a new dyadic chain, F(S, T=that):
 Ɱ - for each character, t, in T:
 f - (S) filter keep any of (the single character, t)
 ḟ - (S) filter discard any of (the characters in T)
 ; - concatenate

JavaScript (Node.js), 60 bytes

f=_=>_.sort((a,b)=>(p=g=>-~`f=$f`.indexOf(g)||51)(a)-p(b))
f= Function Name
 _=> Input _
 _.sort((a,b)=>( )(a)-p(b) Sort By 
 p=g=>-~`f=$f`.indexOf(g)||70 Helper function which takes the index (in the function as string)
 of each char in the input, if it doesnt match any character
 then is assigned a value bigger than the max length of the function

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Jelly, 16 14 bytes

“Ṿv`Ṿ;³³i@Þ”v`

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A full program that takes a single argument, the string to be sorted.

Thanks to @JonathanAllan for pointing out a bug!

Explanation

“Ṿv`Ṿ;³³i@Þ” | The string "Ṿv`Ṿ;³³i@Þ", referred to below as S
 v` | Evaluate this string as Jelly code using the string itself as the argument

The string above evaluates as:

Ṿ | Uneval the string S (effectively wraps it in “”)
 v` | Eval it (effectively removes the “”)
 Ṿ | Uneval it again, adding back in the “”
 ;³ | Concatenate the input to this
 ³i@Þ | Sort the input with respect to the first occurence of each character in the “”-wrapped string S concatenated to the input

Ṿv` is a no-op effectively here, but exists to ensure all characters are represented.

Python 2, 102 100 96 85 79 76 68 bytes

s="print sorted(input(),key=lambda x:('s=%r;c'%s+x).find(x))";exec s

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-2 bytes by using this

-4 bytes by realizing that <0 == ==-1 and removing the unnecessary +1

-11 bytes thanks to Neil

-6 bytes thanks to dzaima

-3 bytes thanks to rod

-8 bytes thanks to negative seven pointing out that the program can output a list of chars

Charcoal, 37 bytes

≔´≔´´´η´Ｆ´Φ´θ´⁼´ι´κ´¬´№ηＦηΦθ⁼ικΦθ¬№ηι

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≔´≔´´´η´Ｆ´Φ´θ´⁼´ι´κ´¬´№η

There are two ways of quoting characters in Charcoal; ´ quotes any single character while ”y...” quotes any character except ” and also counts as a separate string. It turns out that the overhead in having to deal with ” means that it doesn't end up any golfier.

ＦηΦθ⁼ικ

Loop over the characters in turn, outputting any matching characters from the input. This sorts the input.

Φθ¬№ηι

Output any unmatched characters in the input.

Ruby, 57 bytes

->ss.sort_by%->s.ort_byindex()9.index(o)

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Fairly straightforward, assuming I haven't missed a golfing trick. Take in a list of characters and sort by their index in a string consisting of all the uniq characters in the code in order of their appearance. Often their first appearance is in that very string, but that doesn't change the order.