Thanks for the answers. That was similar to what I thought (since the external servers grab their data from core nodes which are full clients).
Currently there is some discussion about hard forking Bitcoin so that the maximum block size is increased. That would require a mandatory update for Bitcoin Core. But what about alternative clients like Electrum and Multibit? Do these also need to be updated or not? Since they rely on external servers, is it just these servers that need to be updated or everyone who uses the program?
Only if the majority of the miners agree there can be a protocol update. Electrum and Multibit have active development. A block size increase probably has effect on the source code.
If a protocol update was announced and not all miners agreed to the fork then the network would bifurcate. Electrum and Multibit have rapid development as you point out so I would think that if there is any compatibility issue (and there seems to be none for the example that I posted in the OP) then they would modify their code to "speak" with the core nodes on the correct fork of the Bitcoin blockchain.
At least that was my assumption.