Those who insist on using the word 'Convict' to describe certain MPs might like to check a dictionary.
Doing a search for 'convict meaning OED' gets:

convict
verb
/k?n'v?kt/
declare (someone) to be guilty of a criminal offence by the verdict of a jury or the decision of a judge in a court of law.
"the thieves were convicted of the robbery"
Similar:
declare/find/pronounce guilty
sentence
give someone a sentence
send down for
Opposite:
acquit
clear
noun
/'k?nv?kt/
a person found guilty of a criminal offence and serving a sentence of imprisonment.
"two escaped convicts kidnapped them at gunpoint"

Most other dictionaries come up with similar definitions.
This would seem to mean that you cannot become a convict by accepting a FPN (fixed penalty notice) as they are issued by the police with no input from the courts. This also explains why those accepting such notices do not have a 'police record'.