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Telecommute includes commute by way of analogy, just like the presence in telepresence, the desk in a virtual desktop and arguably, the friendship of a facebook friend.

That is, although there is no commuting in the traditional sense, some of the abstract properties still hold. For example, the person is considered to be ‘at work’, with deliverables and accountability.

Although they don’t ‘go’ to the office in the traditional sense, they still ‘go to work’ in a more abstract sense - one that carries consequences if the work assigned is left undone.

It’s a portmanteau. When telecommuting you are commuting via the telecommunications network. All your "travel" is done by the internet. No internet, no work (cf no train, no work).

You’re right in that the word "telecommuting" doesn’t literally mean what it means, but it’s much nicer to say than "telecommunications commuting".

The Oxford English Dictionary describes telecommuting as

The action or fact of working remotely, esp. from home, using
telecommunications technology.

Meaning that the 'commuting' is metaphorically by phone, computer, Remote Desktop Connection etc.

You shouldn't take the word 'commuting' too seriously. Telecommuting is just one of those silly coinages that are supposed to be 'humorous' or whatever.

If you don't have access to the OED, here's an M-W link & definition:

to work at home by the use of an electronic linkup with a central office

You could look a the difference as being that teleworking can be done out of hours at the worker's convenience. For instance the teleworker could be provided with scans of paper invoices and delivery notes and be expected to enter all the information into a spreadsheet by 9:00am the next day. If they chose to do this outside the normal working day, perhaps when their children were in bed, this would not really matter so long as the deadline was met.

With telecommuting there is the expectation that the worker will be available by phone, Skype or other online conferencing facility during normal office hours regardless of their domestic circumstances. If online conferencing was involved there would, almost certainly, be a requirement to adhere to a dress code as well. The teleworker could be working in nightwear or the nude, the boss would never know.

Ultimately the telecommuter has a genuine presence in the workplace, even though that is a distance. The teleworker does not, necessarily, have such a presence.

The issue here seems to be whether the use of a verb telecommute is a sensible way to describe the kind of working that it describes. I am surprised to be writing on this site that whether it is sensible is a question neither of grammar nor of usage. The verb is well-established in the leading dictionaries, with a more-or-less common range of definitions. That means that the lexicographers are satisfied that this word is frequently enough used in a wide enough range of contexts.

With hindsight, it was inevitable that the internet would break many of the remaining controls on the English language. For example, it made it possible overnight to turn nouns like access into verbs, like access: and not just verbs but transitive verbs. Any such ‘rules’ were ‘policed’ by editors, lexicographers and school teachers. We now access the internet and all sorts of apps and opportunities it offers. But this phenomenon is a fact of usage.

The verb to telecommute and its cognate noun telecommuting are fully established. I should add that there seems to me to be a subtlety about their use. I have never heard anyone saying “I am a telecommuter” or “I telecommute”. The words occur discursively, mainly in journals, books and reports related to the discourse of sociology, management, and so-called human relations.

Actual ‘telecommuters’ (and here I have to rely on personal experience) use the term. They more often say that they ‘work from home’. Confusingly, not all people that work from home are telecommuters. Telecommuting implies a base to which you work from home. It is in essence remote working. Moreover, it is not clear that all forms of working ‘from’ home count as telecommuting. For example, for several years I ran a company with an annex office near home, visiting the main office once a week. It is not obvious whether this was a case of ‘telecommuting’ or not. Nor is it clear that by having my PA come to work at my house this would have made any difference.

Commuting itself is a very young idea: less than a hundred years. The term ‘telecommuting’ is still a toddler. Let’s give it a chance to come of age.

English is not always logical, and when it is logical, the logic isn't always what you might naively expect.

In this case, the logic is that commuting is what's most relevant to the word. Physically commuting to work is, for most people, a disliked waste of time & resources. Working remotely eliminates the physical commute, so calling it telecommuting emphasizes the fact that it removes that waste from your life. If you called it telework... Well, you're still doing the same actual work, so it doesn't have the same connotation of advantage.

Commute and commuter have themselves changed meaning in the last hundred years or so; originally (and perfectly rationally) the meaning "make a single large payment to buy off future obligations" was applied to paying for your travel in advance ("There are many business men who practically divide their time between New York and Chicago, and ‘commute’ (the American term for taking season tickets)" Daily Chronicle, 1906). The phrase 'commute every day' would then have made no sense; but over time, people unfamiliar with the word assumed that if commuters were those who travelled to and from work, commute must mean 'make the journey'. If 'commute' can take on another meaning because of the perceived lack of a suitable term, why should not 'telecommute' be coined for another such lack, regardless of etymology?