If you've read that you've not been reading the right things! 
Its not true in short. What's just as important is the support you get and the way in which you are treated, not just the way you give birth.
Its also not how 'textbook' or emergency your previous birth was. You can have a birth which is 'good' on paper but to you can be very traumatic for various reasons.
There is a specialist in birth fear at the hospital I plan to go to. His policy is not to give an ELCS just because a woman has fears. The hospital policy is never to refuse an ELCS - but what they find is lots of women do request them believing the same as you or because they feel its the only way they can be in control/have a calm experience. But for some women it is totally the wrong approach and a VB is appropriate. They just need additional support to do that though. The key is building trust and a relationship to allow women to feel able to do that.
Without knowing the reason for your previous CS and what your anxiety is about, its impossible to say which group you might fall into. But the real thing you need either way is, support.
An ELCS for anxiety isn't the magic solution that its touted to be. It misses the point, that its about listening to a woman, respecting her and support her - and that could mean a VB as much as an ELCS.
I would ask to speak to the consultant asap to give you the option of an ELCS if you want, but I think you need to have a think about exactly what you are traumatised by and what your fear is - get to the root of the fear to understand it, so you know what you need to minimise to prevent it happening again.