The word 心底 has 2 readings which are しんそこ and しんてい. Both seem to be interchangeable with one another. Searching online has this page popping up. https://detail.chiebukuro.yahoo.co.jp/qa/question_detail/q10112707945. To quote it directly, the difference is explained as, "「しんそこ」 は思っている状態、「しんてい」 は思っているそのこと。". So to phrase it differently, しんそこ is used to describe a state of thinking whereas しんてい is used to emphasize a thing about thinking? Forgive me if I got it wrong or misinterpreted it but the answer doesn't really seem to explain the proper usage or reading of the word to me.
You've included a link to Chiebukuro but that site actually links the more helpful kotobank definitions here.
First, both readings are correct. It isn't a case of one or the other being right or wrong. There are times when Japanese words can have the exact same kanji and be read in different ways, with one being considered 'correct' - this is called 慣用読み ("reading of a word or kanji using its customary pronunciation (which is not necessarily correct)" - EDICT). A word like 貼付 would be an example of that.
However, in the case of 心底, it appears that each reading can convey a slightly different semantic nuance. For example, the kotobank definitions list しんそこ as referring to feelings and emotions:
心底から感服する to admire someone from the bottom of your heart. (feelings/emotion)
On the other hand, しんてい seems to refer the actual thoughts that someone has, thoughts that are deeply believed.
心底を見抜く to see through to what someone really thinks in their heart. (thoughts)
In some ways, the distinction is just academic. I would say that there isn't really much difference between the feelings in your heart and the thoughts that arise from those feelings. It seems like a trivial distinction, in my opinion.
In any case, both readings of 心底 are correct. They convey slightly different meanings, at least according to some sources.