score:2
This is an easy answer although it may not be what you want to hear. Unless you are able to get it fixed at a CBP office, stay within the dates on your physical stamp. Out of an abundance of caution I would defer to what is on your passport over what is in their database.
Why? Because immigration officers are specifically instructed to issue six months entry for B visitors unless they have strong reasons not to do so, and in that case they must justify the reason to a supervisor/district director.
Thus it is not trivial when an immigration officer admits you for less than six months. Although in your case given the errors in your actual entry it could possibly be a mistake that he gave you only two months, do not risk it.It is a risk vs reward question.
8 CFR 214.2 - Special requirements for admission, extension, and maintenance of status.
(b)Visitors -
- (2) Minimum six month admissions. Any B-2 visitor who is found otherwise admissible and is issued a Form I-94 (see Β§ 1.4), will be admitted for a minimum period of six months, regardless of whether less time is requested, provided, that any required passport is valid as specified in section 212(a)(26) of the Act. Exceptions to the minimum six month admission may be made only in individual cases upon the specific approval of the district director for good cause.