you have already pointed the way to the correct classification by yourself by naming the basic arenaceous rock "quartz arenite". As glauconite is part of the clast community, I would at the very end call it a glauconitic quartz arenite or glauconite-bearing quartz arenite. It is a pure descriptive way which I prefer to any classification scheme which has any genetic connotation.
If glauconite is not RF (rock fragment) , then as Dr. Dill mentioned, it should be quartz arentie. Although you notice that glauconite is a part of framework, but at least one grain is rounded and looks RF.
for a correct name for this rock, the best way to nomenclature you can use the base classification of Saandstone, of course, after the counting of the main framework components of the rock, as appear in your thin section the main component is quartz, so if the percent of the quartz more than 95% you can name this rock as Quartz arenite, and here the glauconite is a descriptive copmonent , so you can name this rock descriptively as Glauconitic quartz arenite or in general as Glauconitic sandstone.
First thing you have to see is......is it a clast or matrix supported framework and accordingly you can use the term arenite/ wacke.
the next thing is to precisely know the amount of quartz among the clast component.
While making out the percentage you have to calculate within the clasts only....Not with reference to cement/ matrix and total volume percentage of the sandstone.
Sandstones are classified with triangular diagram having three ends of quartz, feldspars and rock fragments.
Accordingly they are classified as quartz arenite/ feldspathic-subfeldspathic arenite and lithic arenite depending upon dominant clast component.
glauconite doesnt fit in here as major clast component to be considered for classification.
But as the presence of this mineral is genetic connotations you can always mention it as glauconite bearing quartz arenite
If glauconite is in clastic form then the rock will be arenite, quartz arenite or if glauconite is more than 5% we can call it glauconitic quartz arenite.