I get the feeling (no pun...) that 考える refers to a more objective action than 思う, which refers to one's subjective thoughts/beliefs.
This is absolutely correct. For example:
思う：そこの銭湯【せんとう】はとてもいいと思う。 - I think that bathhouse is really good.
考える：ゆっくり考【かんが】えればわかる。 - If you think carefully, you'll understand.
Sometimes you'll see sentence where either is appropriate, but the former is indeed a bit more subjective than the latter in cases where they both fit.
Judging from how I've seen the word used in print (admittedly, I'm not a native speaker, so take this with a minor grain of salt), it seems to have the same kind of duality that the terms "way of thinking" and "mindset" have in English—they can include one's subjective views and also one's logical thought process.
For example, in this article it refers to one's logical mindset.
Alternatively, ejje.weblio.jp lists this example sentence:
儒教の価値観が今なおほとんどすべての日本人の考え方に浸透している。
Confucian values still permeate the thinking of virtually the entire Japanese population.
This uses the term 価値観【かちかん】 that you mentioned before. So, it seems that your initial belief was correct, and the term can refer to both.
