I think what is accepted at stores or recycling centers is very much local policy and depends on both regulations (local law) and the way collection and recycling schemes are setup and funded. In the Netherlands where I live for example, all types of packaging plastics including bags, but excluding other plastic objects and biodegradable plastics, are collected via communal collection schemes. The reason why they choose for all packaging plastics is that manufacturers are required by law to collect this.
The types of plastic you are referring to are #2 High-Density PolyEthylene (HDPE) and #4 (Low-Density PolyEthylene (LDPE). HDPE is relatively easy to recycle and is also the most recycled type of plastic. LDPE on the other hand often isn't recycled. How difficult it is to recycle a plastic depends on the resin type, but also on if it's contaminated with dyes and other chemicals.
In order to recycle a plastic properly you need to separate the different types of plastic and remove contaminations as much as possible. I suspect that in your case policy is to gather only 1 specific type of plastic object so separating is less of an issue for the recycling center (to reduce costs). Whether or not you can safely return #4 plastic bags to a shop depends on what the shops do with it and if the plastic types are properly separated.
EDIT: according to this curbside recycling program they don't accept plastic bags because their equipment can't handle it properly. On a different webpage they also say that
Don’t be fooled into thinking that all (or even most) of the plastic collected by those programs is actually recycled! There are very few domestic (U.S.) markets for plastics numbered 3 through 7. China, one of the principal foreign markets for U.S. plastics, has recently undertaken measures to stem the influx of non-recyclable and contaminated materials hitting their shores. At least for now, “the markets for #’s 3-7 plastics have dried up!” ......
The bulk of post consumer plastics #3-7 collected in this country have always gone oversees, primarily to third world countries where environmental controls and worker safety standards are far inferior to those in the U.S.
So my guess is that most #4 plastic, no matter the form is either shipped oversees, dumped on landfills or burned.