That was a very useful summary for someone new to the concept of "climb segments" but I'd like to put a finer point on this one.

Before the crash of AA191 (May 1979), industry standard practices were to maintain V2 through second segment climb (even with engine failure).

But this video is an excellent description of how that past practice didn't account for any flap/slat malfunction.
https://youtu.be/e3lzgrFuM4s?si=B_ha2QpBEmVkGVSI

After this accident, industry practice changed to climbing at V2+10 (or +15) until acceleration height in order to provide additional margin above stall speed in case of flap/slat malfunction.