All this working as it should be. Please look at the Feature Reference doc for Call Center Elite, starting from page 109 in the attached document.
Logical Agent implies that telephones are no longer pre assigned to hunt groups. When the agent logs, the telephone becomes associated with all of the skill hunt groups that are assigned to that agent login ID.
Look at Logical Agent Capability page 127 and on Additional agent log-in ID capabilities page 135.
Login ID name on the telephone display
A call to a logged-in EAS login ID by default displays the name associated with the login ID and not the name that is associated with the telephone. This is also true on the receiving party's display for a call that is made from a telephone with an agent logged in. However, the user can display the name of the physical telephone where the EAS agent is logged in. The user must be active on a call with the agent, and must have a telephone with an alphanumeric display and an inspect button. When the inspect button is pressed during a call to or from the EAS agent, the physical telephone name of the agent is displayed.
Coverage path
Call coverage can occur whether or not the agent is logged in. If the agent is not logged in, the busy criteria is met and the call follows the points on the coverage path. If the agent is logged in but fails to answer, the don't answer criteria is met and the call follows the points on the coverage path. A call to the login ID goes to the coverage path that is assigned to the login ID rather than to the coverage path that is assigned to the telephone extension.
Documentation for ACD:
Page 153
Name:
Calls to the login ID display the name associated with the login ID and not the name
associated with the phone. This is also true for calls made from a phone with an agent logged in,
page 156
Displays - Phone:
When an EAS agent logs in, the display for originators who call the login ID shows the login ID and agent name (as administered using the Agent Login ID form). Calls that the agent originates show the agent login ID and agent name at the receiving telephone display. However, the user can display the name of the physical extension where the EAS agent is logged in. To do this, the user must be active on a call with the agent, and must have a telephone with an alphanumeric display and an inspect button. When the inspect button is
pressed during a call to or from the EAS agent, the physical extension name of the agent is displayed.
Working as per design
P.S. SIP phones do not have support for inspect button as of now. So if Agents are using SIP phones, then there is no way to display extension number instead of Agent ID.