Dependence of primary alcohol oxidation on presence of water and "strong" or "weak" oxidants It is known that the oxidation of primary alcohols can result in aldehydes or carboxylic acids. The product obtained from the reaction is described as dependent on reagents and conditions;[1]


  "Primary alcohols are oxidized to either aldehydes or carboxylic acids, depending on the reagents chosen and the conditions used."  


A distinction is typically made between "weak" (eg. PCC) and "strong" (eg. chromic acid) oxidants[2], the former being capable of giving an aldehyde product while the latter carries out the oxidation to carboxylic acids. However, the accepted mechanism for the reaction shows that the oxidation always results on an aldehyde, which can then be further oxidized to a carboxylic acid through an hemiacetalic aldehyde hydrate forming in the presence of water.[1][3]  


  "Most other commonly used oxidizing agents [...] oxidize primary alcohols directly to carboxylic acids. An aldehyde is involved as an intermediate in this reaction but can’t usually be isolated because it is further oxidized too rapidly."   




This indicates that it is the presence of water in the reaction medium which determines the product of the oxidation. Thus, "weak" oxidants could still yield a carboxylic acid in aqueous solutions (in fact, they are always noted to be used in anhydrous conditions) and "strong" oxidants will result on an aldehyde when used with organic solvants. If so, I fail to understand the classification of "weak" and "strong".
Furthermore, some sources, like Wikipedia, seem to imply the existence of a direct oxidation from primary alcohol to carboxylic acid, but I've been unable to find such a mechanism in bibliography, always ending up with the hydrate intermediary method.
To summarize:


Is the distinction between "strong" and "weak" oxidants meaningful or does the presence of water always determine the reaction product?
Does a method for the direct oxidation of primary alcohols to carboxylic acids without an aldehyde hydrate exist?  


McMurry, John. Organic Chemistry. Brooks Cole, 2011. 8th edition.
Alcohol Oxidation: “Strong” and “Weak” Oxidants
Reagent Friday: Chromic Acid, H2CrO4
After looking up more information and taking the comments of user55119 into account, I've decided to post an answer myself, hoping to clearly address the main question points.  


The determinant factor of the products obtained from an oxidation reaction of primary alcohols is the presence of water in the reaction medium, as it allows the formation of an aldehyde hydrate which can be further oxidized. Thus, the distinction between "weak" and "strong" oxidants is not necessarily informative (and could even be misleading) and emphasis should be placed in the reaction mechanism alongside the reaction medium.  
The oxidation of primary alcohols with Cr(VI) reagents takes place through the mechanism depicted in the question or slight variations of it (depending on the reagent chosen). Carboxylic acids are obtained only through an aldehyde intermediate.


Further reading:
McMurry, John. Organic Chemistry. 8th edition.
MasterOrganicChemistry