Jelly,  19  17 bytes

Ḳµ7ị“*;nÄƲ]³Ṙ»i)Ḍ

A monadic Link accepting a list of characters which yields an integer.

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Pretty much a port of my Python 2 answer.

Previous

ḲŒ¿€i@€“©¥q£½¤MÆÑ‘Ḍ

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There is quite possibly a shorter way, but this is a way that first came to mind.

Python 3, 107, 91, 77, 90 bytes

-16 bytes by Sriotchilism O'Zaic

+13 bytes to remove leading zeroes

lambda s:int(''.join(map(lambda w:str('zeontwthfofisiseeini'.index(w[:2])//2),s.split())))

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Retina 0.8.2, 46 bytes

T`zwuxg`E

on

1

th

3

fi

5

se

7

ni

9

D



^0+B

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T`zwuxg`E

The words zero, two, four, six and eight uniquely contain the letters zwuxg. Transliterate those to the even digits.

on

1

th

3

fi

5

se

7

ni

9

For the odd digits, just match the first two letters of each word individually.

D

Delete all remaining non-digit characters.

^0+B

Delete leading zeros (but leave at least one digit).

Python 2,  71  70 bytes

-1 thanks to ovs (use find in place of index)

lambda s:int(''.join(`'rothuvsein'.find((w*3)[6])`for w in s.split()))

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JavaScript (ES6), 65 bytes

A port of Jonathan Allan's answer is 2 bytes shorter.

s=>s.split` `.map(s=>t+=['rothuvsein'.indexOf((s+s+s)[6])],t=0)|t

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JavaScript (ES6),  70  67 bytes

Saved 3 bytes thanks to @ovs

s=>s.split` `.map(s=>o+='2839016547'[parseInt(s,36)%204%13],o='')|o

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Perl 6, 35 32 bytes

{+uniparse 'SP'~S:g/<</,DIGIT /}

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Explanation

{                              }  # Anonymous block

                S:g/<</,DIGIT /   # Insert ",DIGIT " at

                                  # left word boundaries

           'SP'~  # Prepend 'SP' for space

  uniparse  # Parse list of Unicode names into string

 +  # Convert to integer

PHP, 74 bytes

foreach(explode(' ',$argn)as$w)$n.='793251_8640'[crc32($w)%20%11];echo+$n;

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Tried to get a solution which doesn't copy existing answers. I get cyclic redundancy checksum polynomial of 32-bit lengths (crc32) for each word and then do a mod 20 and mod 11 on it to get mixed up unique values from 0 to 10 (missing 6) for each digit. Then using that unique value I find the actual digit.

| Word  | CRC32      | %20 | %11 | Equivalent digit |

|-------|------------|-----|-----|------------------|

| zero  | 2883514770 | 10  | 10  | 0                |

| one   | 2053932785 | 5   | 5   | 1                |

| two   | 298486374  | 14  | 3   | 2                |

| three | 1187371253 | 13  | 2   | 3                |

| four  | 2428593789 | 9   | 9   | 4                |

| five  | 1018350795 | 15  | 4   | 5                |

| six   | 1125590779 | 19  | 8   | 6                |

| seven | 2522131820 | 0   | 0   | 7                |

| eight | 1711947398 | 18  | 7   | 8                |

| nine  | 2065529981 | 1   | 1   | 9                |

PHP, 74 bytes

foreach(explode(' ',$argn)as$w)$n.=strpos(d07bfe386c,md5($w)[21]);echo+$n;

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Another alternative with same length, takes 22nd character in the md5 of the word (only character which gives a unique value for each word) and then uses that character to map to a digit.

Jelly, 20 18 17 bytes

Ḳ7ị“*;nÄƲ]³Ṙ»iƲ€Ḍ

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-2 bytes from running "rothuvsein" through user202729's string compressor.

-1 byte from stealing Jonathan Allan's zero-free enklact string, and putting it in a marginally differently structured program.

Ḳ                    Split the input on spaces,

              Ʋ€     for each word

             i       find the 1-based index (defaulting to 0)

   “*;nÄƲ]³Ṙ»        in "othuvsein"

 7ị                  of the element at modular index 7,

                Ḍ    and convert from decimal digits to integer.

Clean, 88 bytes

import StdEnv,Text

$s=toInt{#i\n<-split" "s,c<-:"rothuvsein"&i<-['0'..]|c==(n+n+n).[6]}

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Heavily based on Jonathan Allan's answer.

Uses a comprehension for indexing instead of indexOf / elemIndex.

Perl 6, 45 bytes

+*.words>>.&{+(1...*.uniname.comb(.uc))}.chrs

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C++ (gcc), 478 218 142 bytes

-(a lot) thanks to Jo King

int f(string s){char c[]="N02K8>IE;6";int i=0,n=0;while(s[i]){n=n*10-1;while((s[i]^s[i+1])+47!=c[++n%10]);while(s[i++]!=' '&&s[i]);}return n;}

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J, 38 bytes

('be~mjPxw['i.[:u:70+1#.15|3&u:)&>@;:

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