"Traditionally" is generally used in a similar manner to "historically".
Both have to do with the past. "Historically" has to do with the past, as it's root, history, is a record of past events. "Traditionally" has to do with the past, because, a tradition is a repeated action, this previous action must have occurred in the past*.
Now, your definition of "traditionally" includes "according to tradition"; but what is tradition? A quick google search gives the definition:
the transmission of customs or beliefs from generation to generation, or the >fact of being passed on in this way.
As a native speaker of English, I would add that "tradition" can also refer to
the customs or beliefs themselves.
Therefore, the difference between "historically" and "traditionally" is that the former is based upon a factual record and the latter is based on custom and belief.
As a general rule, or in informal settings, "historically" and "traditionally" can be used interchangeably. In more formal settings, "historically" can denote an assertion of the past that can be backed up with evidence, and "traditionally" can be used to denote one that cannot. The line between these can be blurry, especially in cases where passed down information and belief(e.g. tradition) is the historical record (e.g. an oral history).
Now to reply to examples given:
1. "Historically" can be used interchangeably with "historically" here.
In this example, "traditionally" is used over "historically" because the historical record is not as important here as the perception of the high pay. It could be that this is the historical fact(in which case "historically" could be used interchangeably), but this sentence is emphasising a traditional belief among investment bankers (e.g. that Goldman Sachs pays the most)
"Historically" could possibly be used interchangeably with "traditionally" here, but it would shift the focus of the sentence. "Historically" would imply that the way they are seen is due to the Conservative Party's voting record. "Traditionally" is used instead because the way they are seen is based upon the party's reputation instead. (Although the two are interconnected; the party's reputation is based upon the way it votes, and the public comments it makes on the issue).
*There is an exception to this, if someone is "trying to establish a new tradition", in which case they are attempting to create an action that will be repeated.