Interpreting over/underdispersion of data with Poisson GLM in R

Question

I have count data that I have fit a GLM to using the Poisson distribution and the default log link function.

I have run a couple tests for over/under dispersion:

1. Dividing the residual deviance by its degrees-of-freedom returns 1.41. Since this is greater than one, it indicates my data is over-dispersed. However, I do remember reading that over-dispersion is not to be of concern if this ratio is less than 1.5.
2. I have seen another source use pchisq(residual deviance, df, lower.tail = FALSE) test, which returned a non-significant p-value for my data (indicating a lack of evidence against equidispersion)
3. I also ran the dispersiontest function in the AER R package which returned a non-significant result, also indicating a lack of evidence against equidispersion.

So, it seems my model is fine in terms of dispersion...?

Fitting a Quasipoisson GLM in R returns an estimated dispersion parameter of approximately 1.40 (interestingly enough this is very similar to my estimation in Point 1). However, I thought a Quasipoisson GLM is supposed to fix any dispersion issues. How exactly should I interpret this - is it also indicating over-dispersion?

Is implementing a Quasipoisson model here over-kill; should I just keep my Poisson model?