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We provide a new scaling relation between $Y_{\text{SZ}}$, the integrated Sunyaev-Zeldovich signal and $M_{500}^{Y_{\text{X}}}$, the cluster mass derived from X-ray observations, using a sample of clusters from the Planck Early Sunyaev-Zeldovich (ESZ) catalogue observed in X-rays by Chandra, and compare it to the results of the Planck collaboration obtained from XMM-Newton observations of a subsample of the ESZ. We calibrated a mass bias on a subset of the Planck cosmological cluster sample using published weak-lensing data from CCCP and MENeaCS, for the new scaling relation as well as that from the Planck collaboration. We propose a novel method to account for selection effects and find a mass bias of $(1-b)=0.89\pm0.04$ for the Chandra-calibrated scaling relation, and $(1-b)=0.76\pm0.04$ for the XMM-Newton-calibrated scaling relation. We apply the scaling relations we derived to the full Planck cosmological cluster sample and obtain identical cosmological constraints regardless of the X-ray sample used, with $\sigma_8 =0.77\pm0.02$, $\Omega_m=0.31\pm0.02$, and $S_8= \sigma_8 \sqrt{\Omega_m / 0.3}=0.78\pm0.02$. We also provide constraints with a redshift evolution of the scaling relation fitted from the data instead of fixing it to the self-similar value. We find a redshift evolution significantly deviating from the self-similar value, leading to a higher value of $S_8=0.81\pm0.02$. We compare our results to those from various cosmological probes, and find that our $S_8$ constraints are competitive with the tightest constraints from the literature. When assuming a self-similar redshift evolution, our constraints are in agreement with most late-time probes and in tension with constraints from the CMB primary anisotropies. When relaxing the assumption of redshift evolution and fitting it to the data, we find no significant tension with results from either late-time probes or the CMB., Comment: 21 pages, 20 figures, accepted by A&A