Interior trim dimensions which are most pleasing to the eye are derived from classical proportions. Architect Marianne Cusato summarizes appropriate dimensions for casings, crown, and base moldings for various ceiling heights here: https://www.mouldingsone.com/designing-with-mouldings

Thank you all for your feedback. This has been very informative for me.

@PPF. If I choose the baseboard that are 5/8" thick, about 1/16" will stick out past the casing. My other option is to use baseboard that is 3/8" thick, would that be better?

Bigger/taller is always better

"Bigger/taller is always better" I respectfully disagree. Interior trim needs to be properly proportioned for the particular space. Mouldings that are appealing in a room with a 10 ft. ceiling probably won't be in room with an 8 ft. ceiling. The folks at Mouldings One have put together a good guide https://www.mouldingsone.com/designing-with-mouldings/

The profile you have chosen (simulated 2-part base) is indicative of a style with lots of millwork in the house. Therefore, assume you have crown as well, maybe some wainscot or simulated panels somewhere in the house. In that respect, and with 9' ceilings and shoe, the taller base is probably appropriate given limited information.

However, the base profile is mismatched to the standard "Colonial" casing that is existing. But, agree on thicker casings or thinner baseboard, so don't know what route you will take. But note thinner baseboard reduces the choices in profiles. And, if you stay with that base profile, and require using a shoe, the shoe should not be a "Colonial" door stop profile, that is too much and competes with the base.

My other option is to use baseboard that is 3/8" thick

Your other other option is replacing the casing :)

Give some thought on the base at the circled location.

@PPF. The baseboard 5/8" thick does flush with the casing in the area that you circled in red. But in some areas, the baseboard sticks out 1/16" past the casing, the installer said he could cut it at 45° where it meets the casing. Is that ok?

The other option is replacing the casing, but there will be too many of them to replace, foyer, TV room, living room...

@3onthetree. Can you suggest which profile for the shoe molding would match with the "colonial" 5 1/2" baseboard?

Thanks

Not "match" per se, just to not compete nor draw your eye down below the base.

The baseboard 5/8" thick does flush with the casing in the area that you circled in red.

I was thinking about the other end. I'm guessing there is no cabinet panel on the near side, only a wall and the base will turn the corner.

A trim package always looks best when the all the parts work together -- same style, same scale etc.

Ideally, no shoe molding as baseboards are scribed to the floor. Shoe will protrude past the casing and most will not do a return, and instead cut it off at an angle.

Showing a returned shoe, and protruding base that would get beveled on the end.

If you choose to use a shoe molding, like the baseboard, it's best if it dies into the side of the casing.

You could add a back band ( https://www.google.com/search?q=back+band ) to your existing casing to accomplish this.

@PPF. The backband will not work as my casing thickness is less than 3/4".

A back band just needs to be thicker than the casing.

A stock back band like is shown here might need to be modified to fit your casing, but it's easy to do.