Can the electron source be known as the anode in one place and the cathode in another?
The anode is where current flows into the device (that is electrons flow out of device); the cathode is where current leaves the device (that is electrons flow into the device). You can use the ACID mnemonic to help you remember (Anode: Current Into Device).

I think there are two possible sources of confusion:


The direction of electron flow is actually opposite to the direction of current (electrons are negatively charged). 
Electrons flow through both the anode and cathode, the question is what direction? I.e. are the electrons flowing from the circuit into the device or are they flowing from the device into the circuit?


For example when you say electrons flow from the anode, do you mean that they flow into the circuit and out of the device (i.e. cell)? If so you are correct. Note the usage of the terms is consistent in CRTs, within the actual tube (the device) electrons flow out of the cathode, but from the perspective of the circuit driving the tube, the electrons are going into the cathode.

Note that which terminal is the anode and which is the cathode is defined by the direction of current flow not polarity, so in power sources (e.g. discharging batteries) the positive terminal is the cathode, but in power sinks (e.g. charging batteries) the positive terminal is the anode.