I learned those verb forms with e as the second vowel: kotev, noten. And they don’t specifically have “he” as the subject. The same form would work for any masculine singular subject. Likewise, kotevet means not only ‘she writes’ but could also mean 'I write" or ‘you write’ anytime the subject is feminine singular. To tell who you’re talking about, you have to specify the pronoun.
The reason this form works differently is that it isn’t actually a verb conjugation, it’s the active participle, and the verb “to be” is implied. So literally ani kotevet means ‘I am someone who writes’, but in Modern Hebrew it’s used to mean ‘I write’. In Ancient Hebrew they had an actual conjugation for the present tense, but in Modern Hebrew this is taken to be the future. Strictly speaking, Semitic verbs don’t really have tenses. They conjugate for the perfective and imperfective aspects. The perfective corresponds pretty well with the past tense and the imperfective with the present or future, though. So we English speakers get into the lazy habit of talking about Semitic verbs as though they have tenses. Near enough.
But note this difference. Verb conjugations also express person and number. So the verb katav is always masculine singular third person, and the subject "he’ is implied. While katavah expresses feminine singular third person, and by itself means ‘she wrote’, and katavti means ‘I wrote’, katavnu ‘we wrote’, etc. But in the present if you just say kotev, that doesn’t tell you who is writing, because it isn’t a conjugation. Grammatically, it functions as an adjective.
The o in the first syllable of verbs in the present is sometimes shown only with a little dot in the upper left corner (holam haser - ‘defective o’), and then the spelling is identical with the past tense, as you said. But more often, I think, the vowel is written with the letter vav and this is called holam male’ (‘full o’).
The letters heh, yod,, and vav, when used to show long vowels this way, are called matres lectionis ‘mothers of reading’. In Modern Hebrew it’s become conventional to use them to write out vowels in full as much as possible when transliterating non-Hebrew names. When the letter vav is used for the vowel, it effectively distinguishes past verbs from the present construction at a glance.
Question: I learned the name of the tenth letter of the Hebrew alphabet as yod. I missed when they decided to rename it “yud”?