/execute as @a at @s anchored eyes positioned ^ ^ ^1 run particle flame

With this command, flame particles will spawn in front of the player every time.

/execute at @a anchored eyes positioned ^ ^ ^1 run particle flame

But with this command, flame particle will spawn at the feet of player but will still spawn in the direction player is looking.

This is because "positioned ^ ^ ^1" take location context (at @a) as an input and display it properly while "anchored eyes" does not.

Because "anchored eyes" affect the execution location of the command I think it should follow the same behavior as "positioned ^ ^ ^1"