Concentration Measures There are several metrics used to identify market power. Such as ... 

the 'n' concentration ratio (the proportion of overall sales given by the top n firms) and the Herfindhal index.

However, there is also the Price cost margin (Lerner index). 

These indicators may not always agree with one another.

Is there some way we can understand which of them is most relevant as an indicator of market power? For example by examining the local characteristics such as what kind of imperfect competition is present, and consumer demand patterns?
Unfortunately, I'm not sure there's a clear answer to your question, since it so often depends on what the specific needs are for the specific project. Overall, there are a number of different ways to try and measure concentration and market power (as you note in the question). 

That said, there's definitely a lot of interplay between different measures, but often these measures can be reworked to allow for looser (or different) assumptions. For example, one of the more well known merger analysis tools currently used by the DOJ and FTC, the Upward Pricing Pressure test is traditionally linked to Bertrand competition. However, as the creators of the measure note, that link is often overstated, and the test can be used in cases where Cournot competition is assumed. 

Moreover, UPP is closely linked to other measures, such as diversion analysis, for which there are many different tools to use to measure it. One method is to use a survey of consumers, (seen, for example, here,) though that of course comes with drawbacks. Instead, market shares can be used, but so can historiical data from switching patterns. Alternatively, you can look at second choice data and/or price and demand elasticity properties, or more abstractly data from stock-out events.

I'm sorry to get stuck in the weeds of merger analysis there, but I think it's an informative example of how many different ways there are to get at the same underlying ideas. Ultimately, what's "best" really depends on the specifics of your project, the data available, and how well you can defend the choice made.

Let me know if there's something more directly relevant that you'd like addressed, but a lot of the theoretical complementarities can be seen only when one particular interest is isolated and examined.