For the sake of your project, make sure you do as @Marc suggests and follow up with the client.
But, in general terms, for asynchronous or batch type communication
- Inbound message - the flow of the message comes into YOUR system
- Outbound message - the flow of the message flows out of YOUR system.
(obviously the direction of inbound / outbound depends on the point of view of the system or esb / eai hub - system architects on the other side of the interface will see things the other way around)
The word transaction might mean many things:
- ACID Transactions (Single phase, e.g. SQL BEGIN TRAN, or Two Phase e.g. XA/DTC)
- Compensated / Long running transactions (usually in context of a ESB or EAI or BPEL engine)
- The data flowing between systems contains Financial transactions (i.e. business domain context, nothing to do with XA / DTC / SQL transactions at all)