Yes that's what happened. And this is a very common mistake.

A bit simplified, but what happens is that each flight is assigned a departure procedure during startup. That procedure is runway specific and designed to keep traffic clear of other runways so they can have traffic departing from multiple runways at the same time.

Imagine a runway on the left and one on the right, the left runway departure procedures would have an early left turn and the right runway departure procedures would be straight ahead until some altitude and then a right turn.

Now if you depart from the right runway but accidentally select the departure procedure for the left runway, the instruments (and autopilot) would indicate a left turn at about 500ft, right into the path of traffic from the left runway.

This mistake is common when for example a plane is first assigned the left runway and then during taxi changes to the right runway. Or the preflight paperwork includes the left runway departure procedure, but the actual assignment from ATC is the right runway (this was a source of incidents in Amsterdam for a while with some airlines)

If you're really interested, read this incident report via Google Translate, it describes exactly how this type of incident happens: https://www.lvnl.nl/voorvallen/20220415-verlies-van-afstand-...

This is exactly why the takeoff clearances say “RNAV xxx, cleared for takeoff”. It’s a last confirmation, right before takeoff, of which departure procedure to use.