That's because in practice insolvency is not so much a financial concept but an accounting and legal one, and in those domains it refers only to situations in which a corporation cannot meet its financial obligations as they come due. In ordinary circumstances that convention gives corporations some leeway to re-negotiate their obligations to stave off insolvency. But during a liquidity crisis it means that an institution whose book value goes negative (temporarily?—who knows?) will or won't become insolvent in part depending on whether third parties are willing to lend to it to plug the hole that exists in its books at the moment. That means that solvency during such periods is a bit of an artificial thing, depending in part on the of vagaries of the marketplace, as well as the judgement and munificence (or whim, if you'd like) of central bankers and other governmental actors.
> Without fractional reserve banking, there cannot be credit.
This is not true. There can be credit, just not with the simultaneous fiction that creditors retain access to the money they've lent. Bond markets and old-fashioned money-market bank accounts operate without that fiction, for instance.
† — Or for that matter during a classic bank run, which is another form of liquidity crisis. A typical bank operating on fractional reserve is solvent in the sense that over some indefinite future time horizon it should be able to give its depositors their money if they demand it, since the money the bank is owned in loans exceeds the money the bank owes its depositors. But a bank does not enough money in its vaults to pay all its depositors if they all want it back at the same moment; if no third party is willing to lend cash to the bank suffering the run ("provide it liquidity"), then it will become insolvent, no matter what its book value.
I'll agree to the second critique re: the credit in a full-reserve banking case.
This is incorrect. Bank solvency has to do with the assets of the bank, even in markets with commodity (e.g., gold) or representative money (e.g., gold-backed paper). The assets of a bank include loans, the liabilities are the money the bank owes to depositors. There is no reason a bank can't take a gold deposit and loan it out (thereby, "creating" gold).
The purpose of a central bank, at least in orthodox economics, is to loan to solvent banks that are nonetheless cash-poor. Imagine a mismanaged bank, that has loaned too much, and cannot meet the demands of depositors. If the loans + cash are more valuable than the deposits, the central bank will loan to the bank to meet their temporary cash shortage. [0]
This is essentially what happened in the case of AIG. The Fed believed that the value of AIGs assets were greater than its liabilities, and loaned them the money at a penalty rate. The Fed believed that AIG was _solvent_. There were a number of extenuating factors here that I'm glossing over, but that is the underlying point. The reason that Lehman was not saved was that the Fed had substantial reason to believe that the assets (primarily the sub-prime loans) were not worth more than the liabilities, and the Fed will not lend into hole. Lehman was _insolvent_.
Fractional reserve banking, by itself, does not suggest solvency or insolvency. Without fractional reserve banking, there cannot be credit. Sharia banking is an example of full-reserve banking, because interest is prohibited, so there is no incentive to loan. (There are ways Islamic banks get around these prohibitions).
[0] I would suggest looking at Bagehot (1873) for a full description of this idea.