Our Minhag is to be Machmir

.משנכנס אב ממעטין בשמחה According to the Chasam Sofer,(O.Ch. 160) the Halacha of משנכנס אב ממעטין בשמחה refers to only one Halacha, "Not to build a בנין של שמחה in the nine days".(starting ר"ח) All the other restrictions, Halachically doesn't start on Rosh Chodesh Av but starts at the beginning of the week of Tisha B'av. שבוע שחל בו. Ashkenazim start the restrictions from Rosh Chodesh because the Rema says that this is our Minhag ואנו נוהגין להחמיר. The Mechaber in Shulchan Aruch uses the term of אסור only for שבוע שחל בו. Even the Rema who holds the restrictions do start from Rosh Chodesh will usually use the term ואנו נוהגין להחמיר בכל זה מתחילת ר"ח etc. The reason for it is because both the Rema and the Mechaber hold that we don't Pasken like רב in the גמרא who holds Halachically it starts from ר"ח.