How to get sum & latest activity in same result?
This should work:

SELECT 
    Name, 
    COUNT(CASE WHEN Activity = 0 THEN 1 END) AS TOTALActivity_0,
    COUNT(CASE WHEN Activity = 1 THEN 1 END) AS TOTALActivity_1,
    MAX(CASE WHEN Activity = 0 THEN LogTime END) AS LatestActivity_0_Logtime,
    MAX(CASE WHEN Activity = 1 THEN LogTime END) AS LatestActivity_1_Logtime
FROM
    Table_1
GROUP BY
    Name ;


Test in: dbfiddle.uk