ID    color  begin_date    end_date     location
1     red    2017-01-01    2017-01-07   initial
1     green  2017-01-05    2017-01-07   nursing
1     blue   2017-01-07    2017-01-15   rehab
1     red    2017-01-11    2017-01-22   Health
2     red    2017-02-22    2017-02-26   initial
2     green  2017-02-26    2017-02-28   nursing
2     blue   2017-02-26    2017-02-28   rehab
3     red    2017-03-11    2017-03-22   initial
4     red    2017-04-01    2017-04-07   initial
4     green  2017-04-05    2017-04-07   nursing
4     blue   2017-04-10    2017-04-15   Health

ID    first_site
1     rehab
2     nursing
3     home
4     Health

SELECT
OVER( PARTITION ID CASE WHEN end_date[0] = begin_date THEN location
                        WHEN location = 'Health' THEN 'Health'
                        ELSE 'Home' end) AS [first_site]
FROM table

def conditions(x):
    #compare each group first
    val = x.loc[x['begin_date'] == x['end_date'].iloc[0], 'location']
    #if at least one match (not return empty `Series` get first value)
    if not val.empty:
        return val.iloc[0]
    #if value is empty, check if value 'Health' exists within the group
    elif (x['location']  == 'Health').any():
        return 'Health'
    else:
        return 'Home'

final = df.groupby('ID').apply(conditions).reset_index(name='first_site')