Is it true that the internal IP address of a computer stays in the packet header as the source IP when it goes outside of the network?

The topic to search on is something like "How does NAT work?" -- Network Address Translation.

The piece you are missing is that at the gateway router, the router's own external IP address is substituted for the internal IP address and generally a new port number replaces the original port number. The internal IP and port number are not disclosed to the outside.

While the transaction is in flight, and as long as the connection subsequently remains open, the router remembers the association between the outbound port number and the internal IP address and port number.

When the router receives an inbound packet, it uses the port number to back-translate the internal IP and port before sending the packet along.

In this way, a sort of firewall is created: an external host cannot initiate a connection with a given internal (protected) host because it has no way to know which port number connects to the target. Connections have to be initiated by the internal host (but I go on to talk about port forwarding below).

NAT routers can also be configured to do port-forwarding. When a connection requests arrives for one of the forwarded ports, it is translated into a connection request on the configured internal IP address and port.

Thanks heaps for the reply! This is all really interesting.
So what you are saying is I might not have gotten to the part of the course that explains the contents of the frame header when it leaves the internal network.  So the gateway router strips off the internal source IP and port and puts on the router's external IP, leaves the original destination IP and adds a unique source port so it can remember which communication is destined for which computer? When it gets back the reply, it strips off the external IP as destination and puts back the original internal source IP and port?

That's awesome! Thanks for that. It makes heaps more sense now!