Unmelt only part of a column from pandas dataframe

set_index, unstack, then melt

df.set_index(['RecordID', 'DisplayLabel']).Value.unstack().reset_index() \
  .melt(['RecordID', 'Source', 'Test'], var_name='Result', value_name='Value') \
  .sort_values('RecordID').dropna(subset=['Value'])

    RecordID     Source            Test   Result     Value
0          1        Web           Logic  Value 1         S
5          1        Web           Logic  Value 2         I
10         1        Web           Logic  Value 3  Complete
1          2     Person           Voice  Value 1       >20
6          2     Person           Voice  Value 2         P
2          3       Mail             OCR  Value 1         A
7          3       Mail             OCR  Value 2         I
3          4  Dictation  Understandable  Value 1         S
8          4  Dictation  Understandable  Value 2         I
4          5        Web           Logic  Value 1         R
9          5        Web           Logic  Value 2         S

Custom function for groupby

def f(t):
    name, df = t
    d = dict(zip(df['DisplayLabel'], df['Value']))
    source = d.pop('Source')
    test = d.pop('Test')
    result, value = zip(*d.items())

    return pd.DataFrame(
        dict(RecordID=name, Source=source, Test=test, Result=result, Value=value)
    )

pd.concat(map(f, df.groupby('RecordID')))

   RecordID     Source            Test   Result     Value
0         1        Web           Logic  Value 1         S
1         1        Web           Logic  Value 2         I
2         1        Web           Logic  Value 3  Complete
0         2     Person           Voice  Value 1       >20
1         2     Person           Voice  Value 2         P
0         3       Mail             OCR  Value 1         A
1         3       Mail             OCR  Value 2         I
0         4  Dictation  Understandable  Value 1         S
1         4  Dictation  Understandable  Value 2         I
0         5        Web           Logic  Value 1         R
1         5        Web           Logic  Value 2         S

Setup

df = pd.DataFrame(data={
    'RecordID': [1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5],
    'DisplayLabel': [
        'Source', 'Test', 'Value 1', 'Value 2', 'Value 3',
        'Source', 'Test', 'Value 1', 'Value 2',
        'Source', 'Test', 'Value 1', 'Value 2',
        'Source', 'Test', 'Value 1', 'Value 2',
        'Source', 'Test', 'Value 1', 'Value 2'
    ],
    'Value': [
        'Web', 'Logic', 'S', 'I', 'Complete',
        'Person', 'Voice', '>20', 'P',
        'Mail', 'OCR', 'A', 'I',
        'Dictation', 'Understandable', 'S', 'I',
        'Web', 'Logic', 'R', 'S'
    ]
})

We can achieve your result by applying logic and pivotting, we split your data by checking if DisplayLabel contains Value and then we join them back together:

mask = df['DisplayLabel'].str.contains('Value')
df2 = df[~mask].pivot(index='RecordID', columns='DisplayLabel', values='Value')

dfpiv = (
    df[mask].rename(columns={'DisplayLabel':'Result'})
            .set_index('RecordID')
            .join(df2)
            .reset_index()
)
    RecordID   Result     Value     Source            Test
0          1  Value 1         S        Web           Logic
1          1  Value 2         I        Web           Logic
2          1   Value3  Complete        Web           Logic
3          2  Value 1       >20     Person           Voice
4          2  Value 2         P     Person           Voice
5          3  Value 1         A       Mail             OCR
6          3  Value 2         I       Mail             OCR
7          4  Value 1         S  Dictation  Understandable
8          4  Value 2         I  Dictation  Understandable
9          5  Value 1         R        Web           Logic
10         5  Value 2         S        Web           Logic

If you want the exact column order as your example, use DataFrame.reindex:

dfpiv.reindex(columns=['RecordID', 'Source', 'Test', 'Result', 'Value'])

    RecordID     Source            Test   Result     Value
0          1        Web           Logic  Value 1         S
1          1        Web           Logic  Value 2         I
2          1        Web           Logic   Value3  Complete
3          2     Person           Voice  Value 1       >20
4          2     Person           Voice  Value 2         P
5          3       Mail             OCR  Value 1         A
6          3       Mail             OCR  Value 2         I
7          4  Dictation  Understandable  Value 1         S
8          4  Dictation  Understandable  Value 2         I
9          5        Web           Logic  Value 1         R
10         5        Web           Logic  Value 2         S

In detail - step by step:

# mask all rows where "Value" is in column DisplayLabel
mask = df['DisplayLabel'].str.contains('Value')

0     False
1     False
2      True
3      True
4      True
5     False
6     False
7      True
8      True
9     False
10    False
11     True
12     True
13    False
14    False
15     True
16     True
17    False
18    False
19     True
20     True
Name: DisplayLabel, dtype: bool
# select all rows which do NOT have "Value" in DisplayLabel
df[~mask]

    RecordID DisplayLabel           Value
0          1       Source             Web
1          1         Test           Logic
5          2       Source          Person
6          2         Test           Voice
9          3       Source            Mail
10         3         Test             OCR
13         4       Source       Dictation
14         4         Test  Understandable
17         5       Source             Web
18         5         Test           Logic
# pivot the values in DisplayLabel to columns
df2 = df[~mask].pivot(index='RecordID', columns='DisplayLabel', values='Value')

DisplayLabel     Source            Test
RecordID                               
1                   Web           Logic
2                Person           Voice
3                  Mail             OCR
4             Dictation  Understandable
5                   Web           Logic
df[mask].rename(columns={'DisplayLabel':'Result'}) # rename the column DisplayLabel to Result
            .set_index('RecordID')                 # set RecordId as index so we can join df2 
            .join(df2)                             # join df2 back to our dataframe based RecordId
            .reset_index()                         # reset index so we get RecordId back as column

    RecordID   Result     Value     Source            Test
0          1  Value 1         S        Web           Logic
1          1  Value 2         I        Web           Logic
2          1   Value3  Complete        Web           Logic
3          2  Value 1       >20     Person           Voice
4          2  Value 2         P     Person           Voice
5          3  Value 1         A       Mail             OCR
6          3  Value 2         I       Mail             OCR
7          4  Value 1         S  Dictation  Understandable
8          4  Value 2         I  Dictation  Understandable
9          5  Value 1         R        Web           Logic
10         5  Value 2         S        Web           Logic