If the coil physically fits the mounting post it should work. I've mounted coils on just one post because it was to narrow and it worked. I've put solid state coils on old Briggs with points and condenser and it worked. It doesn't hurt to try.
Worked properly or fired & ran ?
And you checked the engine with a strobe light to make sure the spark was happening a the right time
or
It ran for long enough to sell it to some one where a year later the engine siezed because it was out of time and overheated ?
And I find it very hard to believe that a magneto module mounted with a single bolt would stay in place for very long before the force from the magnet dragged it around till it touched the flywheel which usually wrecks the magneto & the flywheel .
And it does hurt to try.
The magneto has the timing control chip embedded within it
Some of these have some advance ( retard actually ) built in some don't.
The chip measures the rising voltage being generated in the coil and when it gets to a threshold it closes the primary side and a power pulse is generated.
Because of things like patient laws , different stroke lengths, different positions of the magnets with respect to the position of the piston they are all slightly different.
Because all of this works by the strength of the magnetic fields the coils have slots in them and the slot accounts for about 10deg of crankcase rotation which you use if you are performance tuning one of these engines , closer = early spark , further away = latter spark .
While it is possible that 2 units from 2 different makers might actually be the same it is also possible that you will win the lottery .
A lot of different coils for the same brand of engine are just different lengths of spark plug wire so will interchange quite happily.
I exploit this to reduce my inventory of B & S magneto coils.
If you can not afford to buy a genuine B & S coil or even a known good cheaper aftermarket coil from Prime Line, Stens , Rotary or Oregon the I doubt you can afford a new piston, & con rod should the engine overheat & sieze or brake a con rod .OR even the extra 50% fuel because the spark is a little late so the engine is underpowered and you have to work a lot longer with the governor opening up the throttle fully all the time .
I keep a box full of old coils taken off engines that are broken down for parts and a lot of the independent workshops do the same.
I use these to repair old engines sold without warranty or for checking & occasionally will sell them for 1/3 the new price as they are used & all I know is they worked when I took them off so there is no warranty .
We call these "poverty packs " along with all of the other used , not quite up to scratch part & they are usually put on pensioners mowers because a lot of them either can not afford new parts or don't want to put brand new parts on lest they die before they get used up .
My advice is to seek out an independent repair workshop , the type that work out of old petrol stations or behind a hardwear shop etc etc .
There are a lot more of us than most would think because the bulk of the big mower shops actually only do warranty work & servicing on the mowers they sell and are not interested in fixing mowers if they can convince you to buy a new one .