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Red quasi-stellar objects (QSOs) are a subset of the luminous end of the cosmic population of active galactic nuclei (AGN), most of which are reddened by intervening dust along the line-of-sight towards their central engines. In recent work from our team, we developed a systematic technique to select red QSOs from the Sloan Digital Sky Survey (SDSS), and demonstrated that they have distinctive radio properties using the Faint Images of the Radio Sky at Twenty centimeters (FIRST) radio survey. Here we expand our study using low-frequency radio data from the LOFAR Two-metre Sky Survey (LoTSS). With the improvement in depth that LoTSS offers, we confirm key results: compared to a control sample of normal "blue" QSOs matched in redshift and accretion power, red QSOs have a higher radio detection rate and a higher incidence of compact radio morphologies. For the first time, we also demonstrate that these differences arise primarily in sources of intermediate radio-loudness: radio-intermediate red QSOs are $\times 3$ more common than typical QSOs, but the excess diminishes among the most radio-loud and the most radio-quiet systems in our study. We develop Monte-Carlo simulations to explore whether differences in star formation could explain these results, and conclude that, while star formation is an important source of low-frequency emission among radio-quiet QSOs, a population of AGN-driven compact radio sources is the most likely cause for the distinct low-frequency radio properties of red QSOs. Our study substantiates the conclusion that fundamental differences must exist between the red and normal blue QSO populations.