Yes, legally and canonically mean the same thing in your question. As I understand the case at least three canons apply in such a circumstance, cc. 1226, 1228, and 1229. (Other canons may apply and I'll speak to that in a moment. First of all, a matter of terminology: The term chapel is now used to refer to a private place used for worship and Mass by one or more physical persons; it is not open to the public. A religious community's prayer place where the space may be used for public worship occasionally with permission of the superior is now called an oratory.)
For a diocesan hermit's prayer space to be established as a canonical chapel the local ordinary must approve. If Mass is to be celebrated there either because the hermit is a priest or because the hermit has a priest come occasionally to say Mass, the bishop must approve this as well. (It is recommended that when a hermit is permitted to reserve the Eucharist in her hermitage she also has Mass said there occasionally.) Finally, under canon 1229 in order for the space to be blessed (consecrated) according to the usual liturgical books as a private chapel the space may not be used for any domestic purpose. Moreover the fixtures within the chapel (or, within the space which does not qualify to be blessed as a chapel under c 1229) --- things like tabernacles, monstrance, ciboria, and so forth are separately consecrated. (This is where other canons may apply.)
In my admittedly limited experience diocesan hermits do not ordinarily have the finances or, therefore, the space to dedicate an entire room to prayer without also using some of the space for other domestic purposes (e.g., sleeping). That means that those I know do not have their spaces blessed (or consecrated) as canonical chapels. (Tabernacles can be separated from the rest of the room by free-standing panels or a relatively fixed partition if one is forced by circumstances to also use the room for domestic purposes. Remember, only the hermit will use this space; it is not open to the public.) I don't think this would create a space which qualifies as a private chapel but it does honor conditions accompanying the right to reserve Eucharist in this space.
However, there is an alternative --- though canonically speaking neither does this constitute the space as a chapel; namely, to have the hermitage blessed as a home is blessed and then have the sacred fixtures or appointments like tabernacles, etc. specially blessed. This means that the space and appointments would be blessed by the bishop or a priest designated to do so. From my experience this would tend to be one's pastor. (Sometimes the bishop will come to the hermitage on the day of perpetual profession and in a post-liturgy consecration bless the prayer space then; but often this cannot be arranged because of the distance between parish church and hermitage, or other complications and constraints.) I think most diocesan hermits have their hermitages blessed in a simple home-blessing rite along with a separate blessing of the appointments.
Others will simply trust God to bless the hermitage in its use. By this I mean hermits will often trust that in the living of their lives, through their prayer, silence, solitude, and so forth, God will bless the hermitage with his presence. They may or may not have the chapel appointments blessed by a priest or the bishop separately from whatever is done for the prayer space. This is especially true if the hermit never has Mass said in the hermitage.
I hope this is helpful.
I hope this is helpful.
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