Yeah, the first part's pretty much correct, sounds like you get it - which is what I was going for, well done! That's one part of why indicators are weak acids/bases.
To be honest, the part you don't quite get relates heavily to buffers (so totally understandable), and was probably as a result of my awful wording (I'll go fix that up). I actually meant what happens if a strong acid/base is added to an indicator (which we now know is a weak acid/base) - sorry! What actually happens with an indicator is that it changes colour when the hydrogen/hydronium concentration changes ever so slightly at a certain point - adding strong acids and bases will alter the equilibrium such that the colour of the solution ends up changing due to the concentration of hydronium changing. This is also why strong acids (and particularly the really strong ones) can't be used as indicators (there's either no detectable colour change, no equilibrium exists, etc.).
In the erroneous case you describe (apologising again), the indicator would just get neutralised, I suppose.