Since your question directly refers to an issue of 'meaning', my answer begins with a reference...
Michael Dummett's excellent paper on Intuitionist logic...
https://sites.unimi.it/zucchi/NuoviFile/Dummett73.pdf
...where you will find this quote (first sentence of the second paragraph)
"Any justification for adopting one logic rather than another as the logic for mathematics must turn on questions of meaning."
As Dummett makes clear, he holds to Wittgenstein's idea that, with regards to language, meaning is use.
Dummett was also a scholar of the works of Frege...the author who most clearly expounded the 'modern' idea of translating 'concepts' into logical structures. In Dummett's paper we find this quote:
"These considerations appear, at first sight, to be reinforced by reflection upon Frege’s dictum, ‘Only in the context of a sentence does a name stand for anything’."
Therefore, the meaning of the word 'Infinity' must be understood from its use, from the context within which we find it.
Since mathematicians make extensive use of the word 'infinity' both explicitly and implicitly, one place to gain some understanding of its meaning would be in mathematics literature. What you will find there is a 'mainstream' explication of infinity via the ideas of Cantor. Set theory is assumed as foundational in much of mathematical literature. You will also find alternative ideas related to infinity that are not in agreement with the Cantor/Weierstrass/Hilbert view of a timeless Platonic realm (that most mathematicians take for granted). Many of these alternatives are related to the word 'infinitesimal'. An excellent reference for an understanding of the 'infinitesimal' is a work by J. L. Bell
https://publish.uwo.ca/~jbell/basic.pdf
In this paper Bell addresses the connection between the ideas 'infinite' and 'continuum'. He makes clear the distinction between a 'punctate' versus a 'non-punctate' view of the continuum. This distinction is significant for any understanding of the word 'infinity'.
As for my own understanding as it applies to your question I offer this:
There is no such thing as a perfectly impermeable boundary in Nature. Boundaries only appear to the bounded. The unbounded by definition 'knows' no boundary. Therein lies infinity. Therefore, 'infinity' like 'true' is essentially and permanently beyond 'definition' by us, the bounded. Any attempt to arrive at a definitive result ends in an undecidable, self-referential conundrum. The best we can offer are pragmatic, provisional proposals justified by their utility for purposes we choose based on cultural values.
For those advocates of Cantor's ideas I offer this question: Given the usual two-dimensional array of binary digits shown in the explication of Cantor's diagonal argument...one has to ask: why is it that the string of binary digits in each row is assumed to be a 'completed infinity' instead of the quite obvious: either both the rows and the columns of binary digits are 'complete' or neither is, and by the diagonal argument clearly neither can be. Therein is infinity, itInfinity cannot be bounded. As far as I am concerned, the argument based on that 2D array is an affront to common sense (and to my eyes). It is a cultural artifact of late 19th century Europe.