As soon as I read "Red Lager" I thought to myself, this guy is probably looking for Vienna Lagers (aka Amber Lagers. For whatever reason beer marketers greatly prefer the term "amber" over "red" when referring to lagers, while using "red" over "amber" when referring to ales eg Red IPA). These are lagers with a slightly sweeter profile than the "standard International pilsner" style churned out by every macrobrewery, due to the addition of Vienna, crystal, and/or caramel malts. Indeed, when looking at the examples you gave on Rate Beer and Beer Advocate, RB considers both to be of the "Amber Lager/Vienna" style, while BA considers Dos Equis to be a Vienna Lager and Yuengling to be an "American Amber/Red Lager". What's the difference? Generally, the "American" prefix at the start of any beer style means that it's hoppier, due to our lovefest with hops (at least compared to the Old World). Also, RB doesn't break amber lagers into American and not varieties.
You can use the category page on Ratebeer or Beer Advocate to browse through more examples of the style. Ratebeer's list is default sorted by rating, which isn't always helpful if you don't live somewhere with a great selection, but you can click on "Count" and it will sort by number of ratings; Beer Advocate's is default sorted by number of reviews.
The two beers you mention are very widely available: On RB, Yuengling is #2 most reviewed and Dos Equis is #7; for rating though neither cracks the top 50. On BA, Dos Equis is the #3 Vienna Lager by reviews and #38 by rating, while Yuengling is the #1 American Amber/Red Lager by reviews and #21 by rating. For some great examples of Vienna/Amber Lagers you might be able to find, Sam Adams Boston Lager is rated #44 on RB and #3 on Beer Advocate (why the big discrepancy? RB users tend to be more heavily biased by the idea that certain breweries aren't "cool", while BA users tend to be more objective).
There is always the BJCP guide, which makes the rules that guide most homebrewing and professional contests such as the GABF. The GABF can have real reputation and financial influence for a brewery, so it requires very precise definitions, but outside of competitions most of the brewing world takes the BJCP with a grain of salt (or is it a grain of barley?). Personally, I prefer the Ratebeer and Beer Advocate's style categorizations. Not that they are perfect either - part of the beauty of craft beer is that it is ever-evolving, and brewer creativity knows no category bounds - but it is more democratic and reflective of actual market tastes.
If you are willing to go a little further afield, you might try a Scottish Ales, like Founders Dirty Bastard (www.ratebeer.com/beer/founders-dirty-bastard-scotch-ale/11498/); a Red IPA (not an official style) such as Troegs Nugget Nectar (www.ratebeer.com/beer/troegs-nugget-nectar-ale/30812/); or even a Bière de Garde, France's only indigenous beer style (these are harder to find, but I bet they have them in Montreal. try www.ratebeer.com/beerstyles/biere-de-garde/58/ or www.beeradvocate.com/beer/style/127/.