How is using Jones reagent different than Potassium dichromate in Sulfuric acid for the oxidation of alcohols? I was taught at an institution I study in that using $\ce{K2Cr2O7}$ in the presence of $\operatorname{dil}.\ce{H2SO4}$ will oxidise $1^{\circ}$ alcohols to aldehydes, and $2^{\circ}$ alcohols to ketones. This is supported in here too.

At the same time, I came across Jones reagent ($\ce{Na2Cr2O7}$ in the presence of $\operatorname{dil}.\ce{H2SO4}$) on Khan Academy. But this says that it is an extremely strong oxidising agent and can oxidise Aldehydes to Carboxylic Acids too (thus giving acids on the oxidation of $1^{\circ}$ alcohols).

I do not understand how the above two reagents have different oxidising abilities. Is my premise itself wrong or is there any reason for it to have different oxidising abilities?
Sodium or potassium dichromate or chromium trioxide in dilute aq. sulfuric acid all give rise to Jones reagent. This is used, though less than it used to be due to toxicity and cost issues, to oxidize secondary alcohols to ketones and primary alcohols to carboxylic acids via the aldehyde hydrate. In the case where the aldehyde does NOT form a hydrate with water (e.g. benzylic and allylic aldehydes), then that aldehyde will be stable to the conditions and be the major reaction product. Aliphatic aldehydes will generally give carboxylic acids.

Read more here.