We will begin with some simple background. Omer is the measurement of barley brought as a korban on the second day of Pesach, and it is accepted Rabbinic parlance to refer to that korban as the Korban HaOmer. The mitzva is to count for 49 days, starting on the day on which the Korban HaOmer is offered.[1] There is a machloket regarding whether, in our times, when there is no Korban HaOmer, this mitzva of sefirat ha’omer is Torah law or is Rabbinic.[2]
Regarding your question: Ba’omer almost certainly means “within the period of the omer.” La’omer, might be a different expression of the same idea, but it could also mean “from the time of the offering of the Korban HaOmer.” The Taz[3] assumes that the latter explanation of la’omer is correct. He therefore argues that saying la’omer on the first night does not make sense; the word refers to the count from the bringing of the Korban HaOmer, but on the first night, the count precedes the next day’s omer offering. Accordingly, the Taz maintains that on the first night, one should say ba’omer, and it follows logically that the text throughout the counting should be ba’omer, as the Rama rules.[4]
However, unlike the Taz’s understanding, there is strong reason to suggest that when we use the word omer, we are referring to a time period and not to the actual bringing of the omer. The beracha’s text is “on the counting of the omer.” This makes sense if the omer is a period of time that we count. However, if the word refers to the korban or to the day on which the korban is offered, we are not counting the omer itself, but are rather counting from it. (To deflect this proof, one would have to argue that the usage of this word in the beracha is a slight misnomer.)
The Chok Yaakov[5] demonstrates that the most prevalent text in the time of the Rishonim was apparently la’omer. He supports use of la’omer, explaining that it means from the day of the offering of the omer. He argues that ba’omer does not work well, because it implies that one is counting on one of the days on which the omer is brought, whereas this is appropriate only on the first day.
However, as mentioned above, proponents of ba’omer understand its meaning differently. The Beit Yaakov[6] understands both la’omer and ba’omer as referring to the day within a time period; the question is merely which is the more appropriate prepositional prefix.
Regarding a get and a ketuba, we find that the letter lamed is used for the day number within the month, whereas bet is used for the day number within the week. The Bach[7] presumes that the standard way of counting is to use a lamed, and he suggests a technical reason why the bet is needed regarding the week – so as to avoid confusion. On the other hand, we use phrases such as Tisha B’Av and Tu B’Shevat for days within months, even when there is no concern of confusion.
In practice, perhaps because the Arizal and the Shelah join most Rishonim in promoting la’omer, Sephardim and Nusach Sephard (Chassidic minhag) use la’omer. Perhaps because the Gra joins the Rama to promote ba’omer, many followers of Nusach Ashkenaz use ba’omer. However, the Mishna Berura[8] claims that most poskim choose la’omer, and the Aruch HaShulchan,[9] while citing both texts, prefers la’omer. In practice as well, many people and shuls who otherwise use Nusach Ashkenaz say la’omer. (Interestingly, it seems standard for people to call the most famous day of the omer period Lag BaOmer.) Everyone can and preferably should follow their family minhag.
Realize that the stakes here are very low. Even if one leaves out the word entirely, the counting is valid,[10] and it seems that even if the word would have been needed, both versions are similar enough to be valid.
One’s personal preference is certainly not an excuse to recite out loud a different version than the one accepted in the specific shul in which one is counting (if there is such an accepted practice).[11] All the more so, the chazan or others who recite it for the rest of the community must conform to the community’s minhag.
There are reports of talmidei chachamim who repeat(ed) the count to cover both versions. This is certainly not necessary and probably not preferable (as it is not found in classical poskim). If one is regularly in the practice of covering all halachic bases and wants to do so here as well, we would suggest he do so only unnoticeably.