Since it hit a bird, it was a bird strike. Nothing else matters for that classification.
The puff of flame suggests there probably was a momentary compressor stall. That is the compressor was unable to maintain the pressure as the flow was disrupted by the bird, the flow through the engine slowed down for short while, some extra fuel accumulated in the engine and as the flow restored, the extra fuel burned past the jet, creating that puff of flame.
The engine did not flame out, that is, it continued to run. Older engines tended to get into a repeated cycle of stalls once they stalled as the accumulated fuel from one stall would cause the pressure to increase beyond capability of the compressor and cause another stall, but modern engine computers will automatically restrict the fuel flow to prevent that.
The aircraft continued, because the engine was running and all parameters (RPM, temperature, vibrations, pressure ratio, oil pressure) were indicating normal values.
If the engine stopped, or if some indications were out of tolerance, they would have to return, since two-engined aircraft are required to land as soon as possible when one engine fails. But it clearly did not and they continued (and may not have been aware of the bird strike at all until inspection after landing, because the stall handling is automatic and if no values were out of tolerance, they would not get any warnings).
The engines are designed and tested to continue running after ingesting a small bird , so this was actually the expected outcome. The test usually involves launching dead chicken into the running engine (on the static test bench) from a chicken gun . Said gun is also used for testing canopies and windscreens. Frozen chicken are occasionally used too; contrary to the urban legend not due to stupidity, but intentionally as tougher version of the test .