I mean if you are using social media because you are a councillor it makes sense to have you name begin with "Cllr" or "Councillor" so people know you are a councillor. If it's a unrelated profile, not so much.
I went on a social media training organised my local NALC association and the implication that using the title implies that you are speaking as the Council was never mentioned. It seems a bizzare notion to me.
On the training, the idea of using a seperate Facebook page as a profile for an individual councillor was also talked about, again using "Cllr" wouild be normal for that, it refers to you as a public officer, not the Council itself.
I do sign emails with "Cllr" if it's related to council business and I don't know the person, and I don't think it comes across badly unless someone was to be absurd and call their bank or something and asked to be referred to in that way
As for resolutions about how councillors should use the designation I have no idea how that would be enforced and I hope your council would have better things to discuss.