It's a relative thing. I think it can only be defined relative to some specified alternative, like inflation targeting. It doesn't mean anything otherwise.
There are n different things a central bank could target, where n is an extremely large number. (Strictly, n is infinite, since any combination of two members of n is also a member of n.) One of those n things is the exchange rate. I know what "fixed exchange rates" means: it means the central bank targets the exchange rate. That leaves n-1 other things the central bank could target. But each of those n-1 different things would have different implications for how the exchange rate would change over time in response to various shocks.
If you tell me that a central bank has a "fixed exchange rate", I know what that means. You have picked one out of the n possible things a central bank could be targeting. (Strictly, I would also need to know which exchange rate, and what value, and whether it is a crawling peg or not, but you have still narrowed down the very large set n into a very small subset.) If you tell me the central bank has a "flexible exchange rate" you have told me almost nothing. You have eliminated one of the members of set n, but n-1 is very nearly as large as n. It's like if you asked me where I was going tomorrow, and I told you "I am not going to Toronto". That still leaves a lot of places I could be tomorrow; some of them quite close to Toronto, and some of them a long way away from Toronto, in different directions.
And even that analogy doesn't work exactly, because at least we have an agreed-on measure of "distance from Toronto". Which one is "closer" to fixing the exchange rate: targeting inflation; or targeting NGDP? I don't know. Which one resulted in a smaller variance of the exchange rate would depend on the variances of different shocks, and a lot of other parameters too. And even if they resulted in the same variance of the exchange rate, the covariances between the exchange rate and other variables could be very different.
The Bank of Canada does not have a fixed exchange rate target; it has an inflation target. In that almost trivial sense, Canada has a "flexible exchange rate". Does that mean the Bank of Canada "ignores" the exchange rate? Well, yes and no. When it sets the overnight rate target, eight times a year, it responds to any shock that it thinks might cause inflation to deviate from target unless it responds to that shock. And almost any shock will also affect the exchange rate, and the Bank of Canada will look at every bit of information it has that gives it any clue about the shocks that might cause inflation to deviate from target, and the exchange rate is one of those bits of information. The question is not whether the Bank of Canada responds to unexpected changes in the exchange rate (the "news" it gets at each Fixed Announcement Date) but how it responds, and that will depend on all the other "news" the Bank of Canada gets at the same time.
The Bank of Canada sets a temporary interest rate target 8 times a year, responding to news about the exchange rate and other news, to hit its 2% inflation target. We can imagine an alternate reality in which the Bank of Canada sets a temporary exchange rate target path 8 times a year, responding to news about interest rates and other news, to hit its 2% inflation target. Unless we are inside one of those 6.5 week periods of the very short run, I don't see any way an outside observer, looking at the data, could tell the difference between those two realities. But some US Senators would probably see it very differently. The Bank of Canada would be "managing the exchange rate" in that alternate reality. Or maybe even "manipulating the exchange rate".
The Bank of Canada does not "intervene" in foreign exchange markets by buying or selling foreign exchange (though it reserves the right to do so if it figures it needs to to prevent "disruptive short term movements"). (And just to make the symbolism clear, Canada's foreign exchange reserves are "owned" by the Department of Finance, rather than by the Bank of Canada.) But its words and actions affect the exchange rate just as much as if it did directly "intervene". Such effects are visible to the naked eye in "event studies", because we nearly always see the exchange rate change immediately after the Bank of Canada says or does anything that was unexpected.
I find it very hard to think of any example where the Bank of Canada could do or say anything that would not have some effect on the exchange rate. I find it very hard to think of any example where the Government of Canada could do or say anything that would not have some effect on the exchange rate. In fact, the only examples I can think of where the Bank of Canada and Government of Canada could do or say things without affecting the exchange rate would be if the Bank of Canada had a fixed exchange rate target. The only way to avoid having a "managed exchange rate" would be to have a fixed exchange rate.
(Aside: Hmmm. The only examples I can think of where the US Fed could do or say things without affecting NGDP would be if the Fed had a fixed NGDP target.)
I think this is a sensible answer to what having a "managed exchange rate" means:
Suppose the Bank of Canada changed its policy. Suppose it targeted a weighted average of the inflation rate and the exchange rate. The current policy has the weights at one and zero. A fixed exchange rate has the weights at zero and one. A hypothetical Bank of Canada, with (say) 50-50 weights, would have a "more managed exchange rate" than the real Bank of Canada. The exchange rate would vary less, and the inflation rate would vary more. The bigger the weight on the exchange rate, and the smaller the weight on inflation, the closer it is to a fixed exchange rate, and the more the Bank of Canada is "managing" the exchange rate. That seems like a sensible definition.
We can only define a "managed exchange rate" relative to some alternative well-specified policy, like inflation targeting.
I don't think we can say anything sensible about whether, (say) NGDP targeting is "closer" to a fixed exchange rate than (say) inflation targeting. Whether one has a higher or lower variance of the exchange rate than the other would depend on the type of shocks hitting the economy, and on a lot of things. A central bank which revalued the exchange rate whenever the President went shopping in London might have a very high variance of the exchange rate, but would be "managing" the exchange rate nevertheless.
Just some random thoughts on reading this Vox post by Atish R Ghosh, Jonathan D Ostry, and Mahvash Saeed Qureshi (HT Mark Thoma).