Translation:I can read my email on my cell phone.
Not necessarily. According to Larousse, portable n.m. can refer to any of these: Appareil, machine portable; Ordinateur portable; Téléphone mobile. Technically, other digital products such as walkman/CD/MP3/VCD/DVD players, bluetooth speakers/earpieces/mics, PDAs, tablets, wearable products such as smart watches and VR/AR equipment, etc. are 'portable devices' as well as laptops and cell phones.
It think the confusion lies in how one uses the term "portable". All of those electronic devices that you list out are definitely "portables", but when the term "portable" is used without context, it is usually assumed to be a cell phone.
A similar, but not quite equivalent, example in English is the British use of "mobile" to refer to a cellular phone. Cell phones, tablets, and laptops are all mobile devices, but if a Brit tells you he lost his mobile, he's talking about his phone.
« Portable » in French without context also often refers to a laptop computer. However, the prevalence of mobile phones has progressively shifted usage towards the meaning of a mobile phone, as you said.
There are regional variations, too. In Belgium, for example, a mobile phone is usually referred to as a « GSM », which is seldom the case in France.
Portable is not an equivalent of mobile. In English, portable as a noun is something such as a television, radio, or computer which can be easily carried or moved, whereas a mobile is the same as a mobile phone, according to Collins COBUILD Advanced English Dictionary. Per context (read my email) this word couldn't have had so much ambiguity before smart portable devices became widespread.
I know that mobile was not an exact match, but the idea remains the same. When referring to "un portable" the object in question could be any electronic device in the category, but when referring to a specific portable, without any other context, "mon portable" is most likely to be a cellular phone.
In this specific instance, it would make most sense for it to be a phone(since reading your email on a computer/laptop is not that big of a deal). It could be a tablet, but "cell phone/mobile" is still the most likely translation.
I'm sorry, I'm really not trying to be argumentative, I can just see the reasoning behind the question's acceptable answers. Tablet, is a possible translation, but I don't think it is a likely one. Laptop is more likely, but still, in this sentence, seemingly taken from a conversation, it is most likely a cellular phone. The few French people I talk to regularly at all, say it would be a cell phone 95% of the time.
That would depend how you define "correct". "Courriel" is from a group of words created in Quebec the 1990s to "encourage" people to protect their French cultural heritage by not using the technology terms from the other languages.
The term "Email" has been around since the seventies and is still commonly used. The French and Quebec governments have tried to get rid of it, but just like «weekend» , «email» is still a french word.
So, courriel is "the only correct term" if you work for or care about the opinions of L'Académie Française, L'Office québécois de la langue française or the government's in France or Quebec. But no one in any French speaking country that I know of is going to misunderstand you if you say «email» or the related term «mél» (which is short for messagerie électronique but sounds similar to both "mail" and «tél» in French).