As mentioned in the comments in the current context: "the degrees of freedom for the COVB estimate" and "degrees of freedom for error" refer to the same thing. It is somewhat unfortunate that the documentation of the two functions refers to the same thing with slightly different terms.
As in most cases the term "degrees of freedom" is related to the error/residuals. As this error is directly associated to the $\beta$ estimates of the model used the wording "the degrees of freedom for the COVB estimate" is valid while unfortunately slightly contrived. Nevertheless in both cases what is taking place is an $F$-test about the relevancy of the model examined. You get the associated $p$-value of the upper tail of the $F$-cumulative distribution function. This thread on the mechanics of the General linear hypothesis test statistic gives a nice theoretical overview.
linhyptest is a bit of a niche function. I think bootstrapping your model to get confidence intervals will be far more informative than an asymptotic test based on normality assumption for the parameter estimates. As usually an $F$-test tests the null hypothesis that all the $\beta$ is your model are equal to 0 (aside $\beta_0$); this is a more restrictive assumption and I am not sure it is terribly informative in most cases unless you start defining certain complex grouping among your covariates.
Particular to the MATLAB aspect of your question I would suggest using the standard
regstats function instead of
glmfit given you are not using a particular link-function for your GLM. It will be more computationally efficient as well as more straightforward to use with