The bug
The visual indicator used to state the world is currently being saved is different across editions: Java Edition uses the text "Saving world", whereas Bedrock Edition uses an animation with a chest.
As the indicator was reintroduced to Java Edition in 1.18 after being absent since 1.2.5 (MC-73300), this should count as a valid parity issue.
How to fix
Add Bedrock Edition's saving world sprite to Java Edition.
Further notes
It may be a good idea to have this as a toggle, as to permit switching between the two different types of saving indicator, much like how the attack indicator can be toggled between the crosshair and hotbar versions.
Furthermore, the indicator appears at the bottom right in Java Edition and at the top right in Bedrock Edition. Although Legacy Console Edition is unsupported, it's worth mentioning that the indicator employs a 3D icon on Legacy Console Edition, unlike the 2D icon used on Bedrock. Personally, I believe it should utilize the 3D icon, but ultimately Mojang will decide. If this is addressed, it's likely that a resource pack could enable the use of the 3D icon, similar to Legacy Console Edition.