Eh, I think the bar for tertiary should be pretty low. That's the point of having the four main classification levels :) it allows the more important local roads such as these to be classified as more important than the surrounding purely residential streets. 

I like to classify by a mix of function and form. By function I mean do the roads serve a useful connection in the road network or high traffic volumes. By form I mean do the roads actually support carrying a high volume of traffic or appear as more major roads.

To take an example of yours, Vine Street west of Murray Boulevard is clearly more important than the surrounding residential streets, as it provides access from the main roads into the neighborhood and serves as one of the two main routes to the elementary school there. In addition, it is striped with a middle line and is clearly wider/meant to serve more traffic than the surrounding neighborhood streets. On the balance of those two criteria, I'd consider it tertiary.

Side note, I'd also consider Murray Blvd in this area as a pretty clear secondary road, as it provides an important N/S connection in this area that generally lacks good N/S routes and is more important than all the surrounding tertiary roads. I know it used to be tertiary but it looks like someone downgraded it recently (including to unclassified north of 4500 for some reason; a classification that doesn't make any sense at all given the road's form and function).