Ground from starter runs to frame ground from battery runs to frame is that right

There are too many to count.
Starting in about 1985, when cars & trucks began to have plastic and fibreglas parts, separate ground wires ran from every electrical part to the "universal" ground wires, that ran all over the vehicle.
Have a knowledgeable mechanic show you these.
God bless your efforts.

Just as a test touch a pair of pliers handles across the two big lugs on the solenoid to see if by passing it fixes the problem .The solenoid is bolted to the steel of the tractor for a ground, if not and bolted to plastic there will be a second little wire on the solenoid that grounds it, if it is not grounded or has become a bad connection it will not pass battery to the starter.Inside of that solenoid there is two brass buttons and when you turn the key to start position it send currant through the little wire to make a magnetic field inside of it and pushes away a striker bar that lands across the two brass buttons, with time the arcing make the surface on the two buttons bad.
For the starter to work it needs to be grounded or attached to the negative of the battery in some way. Most mowers this is done by bolting it to a steel frame and also bolting the negative of the battery to the steel frame with a cable. Clip your test light to a cleaned area of the frame,now with the pointed end of the test light touch the positive of the battery to see if your light is grounded at the cleaned frame area, if it lights that tells us the ground cable to frame is good,if not go unbolt the end attached to the frame and sand all surfaces clean and reattach and retest.
Now one more quick test with a test light, clip to the ground lead of the test light to the cleaned frame area again, touch the point end to the engine block, turn key into start/crank position, does it light? If it does it indicates the bottom of the aluminum engine has become oxidized like a dirty battery cable does. As a quick fix you can loosen all the engine bolts and twist the engine left and right in hopes of breaking up the crud then tighten them back.
Another quick test is with a set of jumper cables off you car, neg to the engine block on mower and touch the bolt on the side of the starter on the mower with the positive jumper cable, If that starts it you know your starter is good. Inside of the starter is two magnets glued to the sides of the outside steel case, you hit it with a hammer it can and will break them lose and they slide around till they touch the other magnet. I have JB Welded them back if they are not destroyed.

On most systems, red is positive, and black is negative/ground. An ohmmeter could verify the black wire is ground by showing continuity to the frame or engine. It appears you have two black on the right and one red on the left. You can blow up the battery if you run one ground to positive and one to negative; running an extra to negative won't hurt, it just won't let some part of the system operate, probably the electric starter. If that's an extra red wire in the top center, put both red to positive terminal (+) and both black to negative (-).

Depending on the make model and year, the ground either runs from the battery to the block, or from the battery to the frame, then a strap connects the frame to the engine.

Loose ground wires can cause all kinds of problems, and since ground wires are located in multiple locations specific to each model, each wire must be checked to assure connection.
If you have a starter problem, check the positive connection to the starter as well.
Remove seat and side covers and look closely at the frame to find black and/or black with another color attached to the frame. Remove, clean and reinstall each connection. Paint, rust and other corrosion can degrade connections. Also check your wires for corrosion, especially the battery wires.

Clean and tighten battery cable connections and check battery ground, engine ground-should be a ground strap from engine to the body or frame. If this happens and you have nothing, no headlights, nothing-it's almost certainly a problem at or right off the battery.

The first thing to do is check all of your grounds. You should have solid ground points from the battery to the frame, frame to engine and engine to body. If a ground is bad, it can cause a problem with no charge or even overcharging. Ensure that the alternator is actually providing power using a voltmeter. I've seen rebuilt alternators bad right out of the box.

Did you check the starter? How about the ground - is the battery ground clean and solid to the engine or frame? If you remove the starter, can you make it run with jumper cables?

Just a few things you can take a look at.

You can bolt the ground wire anywhere to the metal body or frame of the car since the battery is attached to the frame and body all electricity flowing through the body/frame goes to the negative terminal of the battery(ground). Preferably you would bolt it as close as possible to the starter without putting in danger of moving parts or heat though.

You have a bad ground wire. check all ground wire locations around engine to frame, and battery to frame.