Question: Is there a precise definition of weakly informative prior?

How is it different from a subjective prior with broad support?

Answer: 
The above comment is accurate. For a quantitive discussion, there are a number of "uninformative" priors in the literature. See for example Jeffreys' prior; see earlier post [What is an "uninformative prior"? Can we ever have one with truly no information?](https://stats.stackexchange.com/questions/20520/what-is-an-uninformative-prior-can-we-ever-have-one-with-truly-no-information/20751#20751)

They are defined in different ways, but the key is that they do not place too much probability in any particular interval (and hence favor those values) with the uniform distribution being a canonical example. The idea is to let the data determine where the mode is.
Further to Eupraxis1981's discussion of informative priors, you can think of the "information" in a prior as inversely proportional to its variance. Consider a prior with near zero variance: you're basically saying "before looking at the data, I'm almost positive I already know the location of the true value of the statistic." Conversely, if you set a really wide variance, you're saying "without looking at the data, I have really no assumptions about the true value of the parameter. It could be pretty much anywhere, and I won't be that surprised. I've got hunch it's probably near the mode of my prior, but if it turns out to be far from the mode I won't actually be surprised."

Uninformative priors are attempts to bring no prior assumptions into your analysis (how successful they are is open to debate). But it's entirely possible and sometimes useful for a prior to be only "weakly" informative.