First off, I don't want to discuss this from the perspective of flexibility as a value in itself, or a goal unto itself etc. That's just silly buzzword nonsense. I'm taking flexibility as a concept here, a thing of life, neither good nor bad, save perhaps in subjective, situation-driven judgement.
(People use the 'flexibility' as a cudgel to get what they want)
We don't have to be flexible, and especially not because an agency or client says that we aren't, as if we should be ashamed of ourselves or feel guilty and, more importantly, give them what they want. Let's make no mistake: they sometimes ask you to be flexible by… accepting they own inflexible demands, for example refusing to sign a non-editable 10-page contract written by their lawyer and full of language that benefits them over you. But it's you who are inflexible and making problems when you won't sign. Seriously?
(All the time, we all do, one just needs to understand this and react properly)
Obviously, we tend to already know something's off in cases like the above, but there are also a lot of cases that, while not with such extreme results, still follow similar basic logic. People stick labels on you when they aren't getting what they want. We all do. This is simply the inherent human subjectivity, the proverbial eye of the beholder. And one more potential accusation that you need to not accept as objective truth the moment a client or agency throws it at you.
- What is that about? What do they want you to be flexible about?
- Do you have any valid obligation to actually be 'flexible about that'?
- What would be reasonable in the circumstances?
- Is their own conduct reasonable?
- Are they flexible to or just want you to accept all their demands unchanged, with zero room for compromise or negotiation?
After answering these questions to oneself, the decision should be much easier — and look much different too than the initial impression or the other party's perspective. (Where the other party's perspective is something that's both morally right and useful to consider, but it simply isn't automatically binding on you.)
(But it's still useful to be flexible)
In my experience it's useful to remain open-minded, suspend judgement and not make one's individual initial preference into an absolute. Frankly, sometimes other people's suggestions are better. Sometimes it doesn't really matter to you, whereas it matters to 'them', hence it would actually be unreasonable not to go out of one's way a bit for a fellow human being. And in some cases you could lose a prospective contract over some minutiae that aren't worth making a stand for.
… The point is to approach this all with a level head and work out an informed decision based on a healthy appreciation of the circumstances and not pressure (intimidation, manipulation, guilt trips, crocodile tears etc.) from the counterparty.
(The point again)
It's perhaps quite worth spelling it out that flexibility certainly does not mean meeting the other party's requests, let alone demands, let alone onerous on unreasonable demands, in full. There is nothing so all-out in flexibility. Flexibility is more about meeting them halfway.
Speaking of which, halfway is a good, reasonable, fast tie-breaker — you split the difference (or the bargain) and cut time losses on the squabbling. It should be employed in business dispute resolution far more often. But, it doesn't work in ethical matters, such as when you're asked to introduce an error to your translation (or legal work, if you're a lawyer, but lawyers generally know this better) and isn't so simple when it's just poor practice either (e.g. sloppy style but preferred by someone who just wants to impress a personal mark on your translation).
But it's perfectly okay to not be completely rigid about fees and deadlines, and payment deadlines, and the way stuff is calculated, for example. Never budging on willingly introducing error into your work is not quite on the same level as never budging on the application of standard billing methods to non-standard situations, now is it? ;) So see, this is the difference.
If you accepted a number of mistranslations and grammatical errors simply to please an agency's proofreader, you'd be doing it the wrong way, and it would be outright silly if you did it while knowing you could be liable but thinking you still owed it to them somehow to expose yourself to that risk just to please them.
On the other hand if you accepted such a request coming from a client who knew the rules but chose to depart from prescriptive correctness for the sake of his target demographic or his own natural experience, that would be the sort of situation where there are pros and cons, all outcomes are subjective, and there are hardly cookie-cutter solutions.
Finally, if you turned down an otherwise highly attractive contract just to avoid departing from your usual billing rules, such as zero discounts for numbers in accounting documents, then chances are you just might be harming your income stream for no good reason.
Okay, one last thing: If there is just one part of this post I'd like everyone to remember, it's that flexibility is on the one hand not capitulating to the other party's inflexible demands just because asked, dared, strongarmed or guilt-tripped to, and on the other hand it can be a good thing for you when it allows you to avoid taking unnecessary losses because of undue attachment to things that aren't really important. Common sense, basically.