You need to change the pin name of the symbol to the desired net name. Symbol, device names are less important here. Here's more info (pasted directly from Eagle 7.2 manual, sect.8.5):
8.5 Supply Symbols
Supply symbols, such as might be used in the schematic for ground or VCC,
are defined as Devices without a Package. They are needed for the automatic
wiring of supply nets (see page 127).
The following diagram shows a GND symbol as it is defined in one of the
supplied EAGLE libraries.
Note that when defining your own supply symbols, the pin and the Device
name need to agree.
The pin is defined with direction sup and has the name GND. This specifies
that the Device containing this Symbol is responsible for the automatic
wiring of the GND signal. The text variable for the value (>VALUE) is chosen
for the labeling. The Device also receives the name GND. Thus the label GND
appears in the schematic, since by default EAGLE uses the Device name for
It is very important that the labeling reproduces the pin names, since
otherwise the user will not know which signal is automatically connected.
The pin parameter Visible is therefore set to off, since otherwise the placing,
orientation and size of the pin name would no longer be freely selectable.
Directly labeling with the text GND would also have been possible here. With
the chosen solution however, the Symbol can be used in various Devices
(such as for DGND etc.).
I suggest reading the whole section, it's less than 2 pages long.