I am used to water meter being in same room with water heater. But, maybe your heater is in garage and meter is at point closer to street where water enters house? But that does not answer your pipe diameter question.

1/2 inch copper has an OD of 5/8 and 3/4 inch has an OD of 7/8. To figure out the supply line, after you cut out the old heater, turn the water back on and see which pipe has water.

On a side note, it appears the electrical line going to the water heater has a splice in it. All splices in electrical wire must be made in an approved junction box and mounted in a permanently accessible location.

Be sure to add a shutoff on the cold water side as well.

Thanks for the replies. Especially the O.D. of pipes ! That just made my install much easier & cleaner !
Yes, Replacing the wire with flex conduit & there will be a box at the wall containing the only splice.
Yes, also planning a drip pan under the water heater.
Good idea to determine which is supply. The main shut off is right next to the water heater (all is in a utility room with the furnace on the main floor. The garage is behind the wall-possible good place to run the drain off the drip pan.)
Good idea about the shut off for the supply line.
I will post before & after pics cause I know we all like home remodel pictures. :)

Also, if the circuit breaker/fuse panel is not visible from the area of the water heater, add an electrical disconnect.

"My PLAN is to use braided hose with the SharkBite connection."

NOT acceptable. Soldering isn't that hard. Practice a bit and do it the correct way. If a 50ish year old woman can do it in a much more confined space than you have, then you can certainly learn to do it when you have all of that space available.

Nothing wrong with sharkbite. Most codes allow it.

Agree - Sharkbite with braided hose is perfectly acceptable. Just make sure that the pipe is nice and round where you plan to use the Sharkbite