Your calculation is likely wrong. See, you can build a laser in a spectrum not generally emitted by stars, and point directly to another star system. You dont need much power, yes, perhaps some terawatts, but it can be pulsed, total energy would be low.
And getting a low power signal doesnt means its impossible to detect, but that the bandwidth is short. Instead of 200 bits per second, you would get 1 bit per hour. But that´s enough.
The simple truth is, if a clone of the human race existed on proxima centauri, long enough for signals to propagate, we'd have no idea and that's an extremely optimistic scenario.
Why would a race be casually pointing terawatts of laser at every star around them. Tremendous waste of resources for essentially no gain within human scale lifetimes.
OP is saying how would we detect a civilisation unknown to us, and for who we are also unknown. A system such as you describe would only be practical for a civilisation trying to communicate with a known counter-party. Otherwise which stars do you target your laser on, and for how long?
Let’s say the Centaurans developed advanced technology 100 million years ago, would they beam this signal at our star for all 50 million years until we happened to evolve? Why our star system, why not any of millions of others? As this system is so directional each detector could only monitor one star at a time. How many do we build, and which stars do we point them at, and for how long?
For direct communication with an established target this would be fine, but as a broadcast system it’s way too target specific to work at galactic scale.
All beams will broaden and distort though. No matter the wavelength or pulsing, anything made by not-a-pulsar is likely to get garbled and shifted beyond comprehension. You can fix this with some really really big fancy lenses, maaaaaybe. But you'd have to know the distance of the point you're trying to image the laser onto first. Collimated light over those distances just isn't ever going to stay that way.
And getting a low power signal doesnt means its impossible to detect, but that the bandwidth is short. Instead of 200 bits per second, you would get 1 bit per hour. But that´s enough.