"un enseignant / une enseignante" is a teacher, ie someone whose job is to teach (anything to anyone).
"un prof" is the colloquial word for "un professeur" (both masculine and feminine), ie someone teaching one or two disciplines to students aged 12 to 18 or in universities.
Context will tell you which article you have to use:
he is a teacher = il est professeur/c'est un enseignant ("un" is masculine)
she is a teacher = elle est professeur/c'est une enseignante ("une" is feminine).
he is my son's teacher = c'est l'enseignant de mon fils ("l'" for "le" is masculine)
she is my son's teacher = c'est l'enseignante de mon fils ("l'" for "la" is feminine)
he is the good teacher I know = c'est le bon enseignant que je connais ("le" to translate "the")
she is the good teacher I know = c'est la bonne enseignante que je connais ("la" to translate "the")
You have changed the article and therefore the meaning.
This sentence states what the woman's profession is, not that she is the specific teacher of a given class. If she were specific, the article would be definite, just as in English:
- this woman is a teacher = cette femme est enseignante
- this woman is the teacher = cette femme est l'enseignante
In English, professions are nouns after "to be/become...".
In French, professions are used as adjectives after "être/devenir...".
- je suis enseignant(e), tu es enseignant(e), il est enseignant, elle est enseignante/nous sommes enseignant(e)s/vous êtes enseignant(e)(s), ils sont enseignants, elles sont enseignantes = I am/you are/he is/she is a teacher; we/you/they are teachers.
Yet, with "il/elle, ils/elles", you can use an alternative construction with professions as nouns:
- c'est un enseignant = he is a teacher
- c'est une enseignante = she is a teacher
- ce sont des enseignant(e)s = they are teachers
The distinction between "this" and "that" is usually not necessary in French unless you are comparing "cette chose-ci et cette chose-là" (this thing and that thing).
Only when it comes to time do you have to be precise:
- These days = Ces jours-ci
- In those days = En ce temps-là
- That year = Cette année-là