(2-part post):
All motoring offences (except some minor parking matters) come under criminal law. While there is no statutory definition of a 'criminal record', it's generally regarded that offences which are dealt with by fixed penalties don't create (or add to) a criminal record. However, offences which lead to arrest and/or a court appearance do.
A criminal record is for life. The slate is never completely wiped clean. However, for most purposes, the majority of offences become 'spent' after a 'rehabilitation period'. The length of this period is determined by the sentence imposed by the court. For an offence dealt with solely by a fine (e.g. your public order offence), the rehabilitation period is 5 years. The rehabilitation period for a disqualification for driving ends at the end of the disqualification. (If, though, you were fined at the same time, the 5 year rehabilitation period applies).
So both your offences are now (for most purposes) regarded as 'spent'. You are no longer required to declare them when applying for such things as insurance or most types of employment. (Even if a question is worded as 'Have you ever been convicted of a criminal offence?', you can lawfully answer 'No' without fear of prosecution for deception).