There is no such limitation as described in the last bullet point provided by DiscoveryBenefits.com and there never has been. I suspect that whoever wrote that confused an HSA contribution with something to do with a self-employed health insurance deduction.
The HSA account belongs to you, not to your former employer. As long as the HSA custodian permits you to deposit contributions into the account (I can't imagine that they would not allow you to do so), yes, you can use the existing HSA account to receive your HSA contributions.
