Are monastics allowed to wear lay disciple's clothing?

Namo tassa bhagavato arahato sammāsambuddhassa (x3) In regards to the spreading of buddhist monasteries in the west within climates being rather different than that in Asia, I've seen pictures of monks wearing jackets, beanies and shoes considering how the weather may be quite cold/wet (in the UK for instance). Yet, isn't an offence in the vinaya for monastics to be wearing what could be considered a lay disciple's clothing ? > "But cold, lord, is the winter night. The 'Between-the-Eights' is a > time of snowfall. Hard is the ground trampled by cattle hooves. Thin > is the spread of leaves. Sparse are the leaves in the trees. Thin are > your ochre robes. And cold blows the Verambha wind. Yet still the > Blessed One says, 'Yes, young man. I have slept in ease. Of those in > the world who sleep in ease, I am one.'" https://www.accesstoinsight.org/tipitaka/an/an03/an03.034.than.html I'd gladly learn more regarding these rules from those wishing and willing to share, May you be well.