The error in the diagram is the separate "DC in" and "DC charge" connections to the inverter/charger.
There is only one DC connection on an inverter/charger.*
There are two AC connections, "AC in" and "AC out", but DC has only one connection - "battery".
As I said, the DC connection can be either...
In (drawing power from battery and suppling power to AC out (operating as inverter))
or
Out (drawing power from AC in and supplying pass through power to AC out and to battery (operating as charger))
...but not both at the same time.
So as Verstad just more or less said - the diodes and manual transfer switch won't work.
Hence Orton's dual inline inverter setup - one "inverter only" (fed by engine battery) to supply AC to a second "inverter/charger" so the inverter/charger can operate in battery charger mode to charge a house battery.
But that dual inline inverter setup is a bit goofy if you ask me. The rationale behind that setup is based on a couple things, one of which is outright false.
The first reason behind Orton's dual inline inverter setup - which he mentions on his page - is that you aren't supposed to mix different lead-acid batteries when charging. This is false. Lead-acid batteries have to be identical (type, size, brand, age, etc.) if tied into a full-time battery bank. If only tied part-time during charging, they don't.
The second main reason for that setup is so that the charger section of the second inverter can do a better job of charging the house battery than simply charging the house battery from the alternator via ACR or split-charge relay. Well, that's a good reason. However, I don't think it's a good enough reason to justify the added complexity. The "alternator->engine battery->relay or ACR->house battery" setup is simple, reliable, proven and automatic, and will get the house battery charged. Eventually. If you drive enough hours.
And there's the rub. One of the reasons for Orton's setup - again stated on his page - is to limit the load on the alternator. So you can charge from the alternator and it will take a long time. Or you can feed a good charger from an inverter fed by the alternator to get a more precise charge to the battery, which should take less time. But if you limit it too much, it won't save enough time to make the whole complex rig worth the bother.
Ultimately, Orton has other reasons (supplying various AC loads), which might justify the complexity of the dual inline inverters, but I wouldn't go about it the same way...
If the second inverter (from the house battery) is big enough to supply the AC loads from the house battery, then I would just rig it to do that all the time.
Then the only issue to be solved is charging the house battery. The most common way to do that is either, A) a split-charge relay, B) Automatic Connection Relay (ACR; split-charge relay with a brain) or, C) B2B (battery to battery) charger.
The B2B will do the most precise job, and cost the most. If it's big enough, it will even do it faster. But lead-acid batteries are not precise electronic devices. They are sloppy chemistry experiments in a plastic box. Great precision in charging normally isn't required.
*[There are of course exceptions to every rule. Some inverter/chargers do have additional DC connections. Commonly they might be for something like "solar in" or "battery bank #2". But none have the ability to both suck battery power and push battery power simultaneously.]