Two English words to distinguish French words “numéro” and “nombre”

Solution 1:

As a software developer, here are a few phrases that I would prefer:

numéro of the current software process

• Version of the current software process. A version number is implicitly part of a version, it is what identifies one version from another. If you are talking about software versions specifically; then this is the best option.
• ID/Identifier of the current software process. This is a more generic programming term, used for any type of identifier.
• If "software process" refers to a Windows process (or similar), you could also call it the handle of a process. A handle is a type of identifier that is used to retrieve Windows processes (or windows), although (since then) the definition has been broadened slightly to also include similar non-Windows system identifiers.

nombre of running software processes

• The (total) amount of running software processes (edit this has been contested in the comments below, I edited my answer to reflect this)
• The (total) count of running software processes (= better to use when communicating with programmers)
• (addition from edit) The (total) number of running software processes

edit The suggestion to use "amount" is heavily contested, as you can see in the comments below. I agree that "amount" is intended to be used with mass nouns rather than countable items, though I do see cases where it sounds better than using "count".

Both "amount" and "count" have issues. They sound a bit weird, and are not as grammatically correct than I thought.
So I'll offer a third suggestion, which takes us back to the beginning of your question:

The (total) number of running software processes

We've come full circle; but looking back on it now; "number" is my preferred option here. It sounds better and is grammatically more correct than "amount".

I would like to mention that, when speaking in the context of programming specifically, "count" is still perfectly valid. I just find that it sounds a bit weird in general English, when compared to "number".

Solution 2:

In software, index or id is used to denote any unique identifier.

The second case is not as well defined, but you can certainly use count.

Solution 3:

To take it out of a coding context, a house number identifies a house uniquely within the context of a street. That's an ID or index number. The number of houses in the street is a count. Just like when writing code you can't assume that max(indices) = count(indices) (in particular the omission of house number 13 is common, as is the existence of houses numbered 7A etc.¹)

This examples serves to demonstrate that in everyday uses we don't distinguish between cardinal and ordinal numbers by word choice. Context, and in the above example word order, do a lot to help.

Of course we can find synonyms that are clearer when we have to: Total/amount/count/quantity for nombre and ID/index/registration number for numéro. These don't generally sound very natural though (that's fine in technical work). In everyday English we'd just say "there are 7 objects, number 6 is the one we're interested in" (back to the house numbers: "there are 14 houses in the street, we want to visit number 15"). As you can see we often avoid using anything equivalent to nombre and use number for numéro.

¹ Zero-based indices further complicated things, and that's all I need to say about them.

Solution 4:

There are no 'short English words to distinguish these two different meanings of “number”' in the way that French has two different words. They simply don't exist. Number is used for both. That's just how it is. Just accept it and learn to know which concept is intended from context.

For the examples you give, you'd usually speak of a process identifier, process id, or PID, but process number would work too.

Current process number: 7494

For "how many" processes are running currently, "count" or "number" could be used:

Number of running processes: 53

or

Process count: 53

It might help you to do a google books search for "number" to review a large number of uses of the word so that you can get more familiar with how we use it.