I mostly agree with this, "
Thameslink" works perfectly for a regional rail network for south-east England focused on cross-London services. It would be a third "mode" of railway in addition to the Underground and Overground networks already run by TfL. As pointed out by @DynamicSpirit they're not perfectly identical in terms of operations, but as far as most of the public are concerned I don't think they'd see any real difference especially if TfL ever took over Thameslink operations and so integrated their tickets, zones etc. If the sub-surface and deep-tube lines can be combined into one "Underground" network then I don't see why the two cross-city heavy-rail lines can't be combined as well.
Still, I do think within a hypothetical "Thameslink network" the lines should be named, just as lines within the Underground network are and the Overground network will be. Then you could keep Elizabeth for the existing E-W line and come up with some new names for the current Thameslink services (though perhaps not quite up to the level of
12 different lines and colours!).