博士 Pronunciation /ʃə/ or /ʃi/
Actually neither of them is correct [technically speaking]. Also in the link you provided, that does sound like /i:/ as in "see", but that is not the sound actually made. Maybe the one provided such pronunciation might put her tongue a bit close to the teeth and create a bit confusion, or the recording is not authentic enough.  

Acceptable way: let the position of your tongue ready for pronouncing /r/ in English. Then to pronounce "sh", just keep this position and let the air out of the mouth. When you do that, the air should pass through the small gap above your tongue.  Now for the full "shì", do the same thing and let your voice box vibrate at the same time. 

In Standard Mandarin or "Putonghua" [普通话], "i" after "s/sh, z/zh, c/ch, r" are never pronounced like "/i/" as in "bit" or "/i:/" as in "see". In these cases, you could think of them as an extension of the sound in the front of "i". 

If someone pronounces "shì" as "shè", then maybe s/he is used to pronounce "shì" for a short time then immediately s/he let his/her mouth open and relax before completing the pronunciation. Technically speaking, they actually pronounced "shì e", but not as much explicit, thus many native speakers would accept such a sound as well. 

The phenomenon "儿化" truly exists, but not all words end by sound "i" are acceptable to  be "儿化" i.e. add a little /ər/ [IPA] at the end. The word 博士 should not be "儿化" in general [although some people may make this mistake because of the influence of their dialect]. 

P.S. "sh" in mandarin is not the same as /ʃ/ in English. 

P.P.S. when referring to IPA symbols, I had used / / to wrap, while to Chinese Pinyin, I had used quotes "" instead. 

UPDATE

For safety, I am here to paste useful explanations I made in comments.  


When I said that 



  you could think of the "i" as an extension of the "s/sh, z/zh, c/ch r" sound in the front of "i" 


I meant that, e.g. when you pronounce "shi", you just need to hold the position of your tongue that produces "sh" and let your air out for a bit of longer time. Equivalently the "shi" sound would sound like prolonging the shhhh sound as @TomSofty33 understood. 


When I said 



  they are actually pronounced "shì e", but not as much explicit


The "not as much explicit" here is referring to the /i/ sound being not to explicitly pronounced as @TomSofty33 understood. 
And the "e" in "shì e" does sound a bit like /ə/ in english IPA as in the word "her".  


The pronunciation in the 1st link provided in the OP does not soun awkward to me as a native Chinese speaker, but might not be clear in the records. Personally I do not recommend. Some people might pronounce 博士 like that in the link. But technically speaking that is not the correct or standard one, however acceptable.
I am a native chinese speaker with satisfying TOEFL grade. So when I write the description starting with an English sound /r/, you could totally do as I explained, you do not need to worry about something like I may pronounce the English words in a wrong way, which would lead you to mistake if you do take what I have written in my answer.