From this answer, I understand that logograms represent words in a language, while ideograms represent concepts independent of a language.
What about numerical digits like "1"?
In the linked answer, these are classified as ideograms, because they represent a concept.
But don't they also represent specific words in English ("one"), as well as other words in other languages ("un", "eins", "uno") and thus can be seen as both logograms and ideograms?
I argue that numerals are true ideograms (or "semantograms", meaning-symbols). Ask yourself: what does the symbol '1' stand for in the following examples?:
In 1 it stands for the word "one", yes. But in 2 it doesn't stand for any particular word, by itself; you have to take it together with another symbol, 0, and read both of them simultaneously as the word "ten". Likewise in 3, with two zeroes, for the word "hundred". What is the symbol "1" itself standing for in these numbers? Well, if you substitute it for a 2 or 3, you'd get from "ten" to "twenty" or "thirty" etc., which are words naming the concepts of one times ten, two times ten, etc. So the commonality between the various uses of 1 isn't the word "one" itself, but rather the mathematical concept of unity.
What word does "1" stand for in example 4, "1st"? Well not a full word, of course; when followed by "st", the entire set represents the word "first", so that the 1 represents the partial sequenc "fir-". "st" acts as a determinant for a special "ordinal" meaning of 1. Similar examples would be things like 1º lugar in Portuguese, where it represents primeir- rather than its normal reading um (incidentally, consider also that 1 stands for the word uno in Italian 1 ragazzo, but for the word una in 1 pizza). 5 show how the same numeral orthography can change 'language' in the same text: the first "100" is read as "hundred" and the second, as cien. The common trend between all those ways of using the symbol "1" is that they're always related to the mathematical concept of unity, regardless of how it's put in words.
Do Chinese characters stand for morphemes, or phonemes?
You can't read a Chinese character without producing a sequence of phonemes. Some have argued that they're just a complicated notation for sequences of phonemes, making them no different than alphabets. However, in their most basic, systematic principle of usage, there's a fundamental difference between phoneme-based notation, and Chinese-style notation: you have to change Chinese characters not only when phonemes change, but also when morphemes change (yīn 'cause' is written as 因, yīn 'sound' as 音, yīn 'flourishing' as 殷…) This is what we mean when we call Chinese characters "morphograms". It's not that they don't represent sounds; they do. It's just that they represent sounds and meanings together, as units; that is, morphemes.
Likewise, an ideogram (or semantogram, semasiogram etc.) isn't a symbol that cannot be put into words and will never represent words, ever. It's a symbol whose usage systematically points to a given concept, which can then be expressed as words according to context (words which, in turn, are expressed as phonemes, just like morphograms/logograms). 1 doesn't systematically denote the word 'one' or any other particular set of words, but the mathematical concept of unity, however it's expressed. (This is unsurprising when we consider that it's not originally a linguistic symbol, but a math symbol.)
If the difference is still confusing, compare it with a Chinese morphogram (or, to keep with your example, let's pick one that works as a logogram, though Chinese symbols are basically morphograms). The Chinese symbol 狗 always represents the word gǒu, 'dog'; so it can be said to be a logogram for gǒu, 'dog'. However, like English with 'dog' and 'hound', Chinese has other words for 'dog', like quǎn; and quǎn isn't written as 狗 but rather 犬. So 狗 cannot in any way be said to represent the idea or concept of 'dog'; it represents the specific morpheme gǒu (which, in turn, is realized as a sequence of three phonemes and a tone). The symbol '1' does not work like that; it represents the concept of 'oneness', no matter if it's realized as the words "one", "ten", "fir(st)", cien…
