To answer your question let's go through the common uses of subjunctive in spanish and try to find a pattern:
- To express feelings: Me gusta que seas trabajador (indicative is allowed to emphasize a fact : Me gusta que eres trabajador)
- To express an opinion: Es triste que no seas trabajador (indicative is allowed to emphasize a fact : Es triste que no eres trabajador)
- To express doubt: No creo que venga (indicative is incorrect: "No creo que viene")
- Wish or hope: ¿Quieres que trabajemos juntos? (indicative is incorrect: "¿Quieres que trabajamos juntos?"). "Que duermas bien" (indicative is incorrect: "Que duermes bien")
- Asking for something or giving advice: Te pido que te lo pienses (indicative is incorrect : "Te pido que te lo piensas")
- Indirect imperative : "Dile que venga" (indicative is incorrect : "Dile que viene")
- To describe something unknown : "Quiero una galleta que tenga chocolate" (indicative is incorrect : "Quiero una galleta que tiene chocolate", but you would use indicative to describe a cookie which is known : "Quiero la galleta que tiene chocolate" )
- To express a goal: "Este dinero es para que te compres algo bonito" (indicative is incorrect : "Este dinero es para que te compras algo bonito")
- To express future: "Cuando venga el paquete lo sabremos". Interestingly enough, when the future is less certain, indicative is used : "Si viene el paquete lo sabremos".
- To express a condition: "Estés o no estés de acuerdo, lo haremos a mi manera"
As you can see, it is extremely difficult to summarize the uses in just a couple of sentences. Most of the time there is a fixed pattern and only subjunctive can be used, these sentences seem to express uncertain, unverified or unspecific facts. When it isn't, indicative is usually used to express more certainty. So the second definition matches better the current use of subjunctive.
In the first definition of 1771 the second paragraph defines an use which is proven wrong by the last example I gave. Subjunctive can be used by its own in Spanish, though rarely.
The explanation about where the word "subjunctive" comes from seems to be correct. It comes from Latin "subjungere" which is the present active infinitive of subjungō that means "join with" or "unite" / "subdue" or "subject"
To summarize, I would say both definitions are mostly correct for current uses, but inaccurate. Still I can't think of a more accurate definition in one