My understanding is that they all mean "up to." But could they mean "up to in different contexts, some in time, for instance, and other with regards to space?
is equivalent to
is used in a temporal context and can be translated with
can be used for
See @Em1's comment for additional cases.
It is not the appropriate way to differentiate between those three words and collocates, respectively. You forget "bis in", "bis nach", "bislang", ...
First of all, depending on its usage the word bis is a different type of word: conjunction, adverb or preposition.
Depending on what you're saying a preposition like auf, zu, nach etc. is following bis.
The word bisher, however, is a compound of bis and her and determine a time range from an indefinite point in the past until now. You can replace bisher with bis jetzt, bis zum jetzigen Zeitpunkt without changing the meaning. If the time span is quite long (i.e. days, months or years) you can also replace bisher with bis heute.
I left out the conjunction since it is not important to your question.