The definition of business under immigration law is rather limited. Generally speaking, a B-1 visa holder is not permitted to work, receive payment from a US source (other than reimbursement for certain expenses), or run a business.
A B1 visa holder may:
B1 in lieu of H1 is referring to a special provision that a B1 or B1/B2 visa holder may be allowed to perform limited, short-term H-1B work in the U.S., if they meet certain requirements:
A person approved for B1 in Lieu of H1 would still be admitted as a B1 visitor, so any criteria governing B1/B2 visas still apply, including the need to demonstrate strong ties to their home country and clear intention to depart the U.S. on time.
Note that B1 in Lieu of H1 has always been somewhat controversial, or "muddy" if you will. Many consulates choose not to issue "B1 in Lieu of H1" and there have been conflicting reports on how USCIS officials would treat such annotations at the POE.